{"id":1360,"date":"2025-03-01T10:25:55","date_gmt":"2025-03-01T10:25:55","guid":{"rendered":"https:\/\/www.mathros.net.ua\/en\/?page_id=1360"},"modified":"2025-03-01T12:33:10","modified_gmt":"2025-03-01T12:33:10","slug":"eulers-method-calculator","status":"publish","type":"page","link":"https:\/\/www.mathros.net.ua\/en\/eulers-method-calculator","title":{"rendered":"Euler&#8217;s Method Calculator with Steps"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"1360\" class=\"elementor elementor-1360\">\n\t\t\t\t<div class=\"elementor-element elementor-element-520b42f e-flex e-con-boxed e-con e-parent\" data-id=\"520b42f\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-f09b256 elementor-widget elementor-widget-html\" data-id=\"f09b256\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"h2-title\">Solve Ordinary Differential Equations the Easy Way\u2014Our Online Calculator\u2019s Got You Covered!<\/h2>\n<style>\n    .h2-title {\n        text-align: center; \/* \u0412\u0438\u0440\u0456\u0432\u043d\u044e\u0432\u0430\u043d\u043d\u044f \u043f\u043e \u0446\u0435\u043d\u0442\u0440\u0443 *\/\n        color: gray; \/* \u0417\u0431\u0435\u0440\u0435\u0436\u0435\u043d\u043d\u044f \u0441\u0456\u0440\u043e\u0433\u043e \u043a\u043e\u043b\u044c\u043e\u0440\u0443 \u0442\u0435\u043a\u0441\u0442\u0443 *\/\n        font-size: 24px; \/* \u0422\u0440\u043e\u0445\u0438 \u0431\u0456\u043b\u044c\u0448\u0438\u0439 \u0440\u043e\u0437\u043c\u0456\u0440 \u0442\u0435\u043a\u0441\u0442\u0443 *\/\n        font-weight: 600; \/* \u041d\u0430\u043f\u0456\u0432\u0436\u0438\u0440\u043d\u0438\u0439 \u0448\u0440\u0438\u0444\u0442 \u0434\u043b\u044f \u0430\u043a\u0446\u0435\u043d\u0442\u0443 *\/\n        line-height: 1.4; \/* 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src=\"mathquill\/lib\/jquery.min.js\"><\/script>\n<script src=\"mathquill\/lib\/mathquill.min.js\"><\/script>\n<script src=\"https:\/\/d3js.org\/d3.v7.min.js\"><\/script> <!-- D3.js library -->\n  \n<div class=\"calculator-container\">\n    \n    <!-- \u041a\u043e\u043d\u0442\u0435\u0439\u043d\u0435\u0440 \u0437 \u0442\u0435\u043c\u043d\u0438\u043c \u0444\u043e\u043d\u043e\u043c -->\n    <div class=\"equation-container\">\n        <label for=\"inputField\">Differential Equation & Exact Solution<\/label>\n        <hr>\n\n        <!-- \u041a\u043b\u0430\u0432\u0456\u0430\u0442\u0443\u0440\u0430 \u0434\u043b\u044f \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u043d\u0438\u0445 \u0441\u0438\u043c\u0432\u043e\u043b\u0456\u0432 -->\n        <div class=\"calcKeypad\">\n            <button class=\"padBtn\" onclick=\"addSymbol('x^n')\">x<sup>n<\/sup><\/button>\n            <button class=\"padBtn\" onclick=\"addSymbol('\\\\sqrt{}')\">\u221ax<\/button>\n            <button class=\"padBtn\" onclick=\"addSymbol('\\\\cdot')\">\u00b7<\/button>\n            <button class=\"padBtn\" onclick=\"addSymbol('\\\\frac{}{}')\">x\/y<\/button>\n            <button class=\"padBtn\" onclick=\"addSymbol('e^x')\">exp<\/button>\n            <button class=\"padBtn\" onclick=\"addSymbol('\\\\sin()')\">sin<\/button>\n            <button class=\"padBtn\" onclick=\"addSymbol('\\\\cos()')\">cos<\/button>\n            <button class=\"padBtn\" onclick=\"addSymbol('\\\\tan()')\">tan<\/button>\n            <button class=\"padBtn\" onclick=\"addSymbol('\\\\log()')\">log<\/button>\n        <\/div>\n\n        <!-- \u041f\u043e\u043b\u044f \u0434\u043b\u044f \u0432\u0432\u0435\u0434\u0435\u043d\u043d\u044f \u0440\u0456\u0432\u043d\u044f\u043d\u044c -->\n        <label for=\"inputField\"><small>Differential Equation: <em>y' = f(x, y)<\/em><\/small><\/label>\n        <span id=\"inputField\" contenteditable=\"true\" class=\"math-input\"><\/span>\n\n        <label for=\"inputField1\"><small>Exact Solution of the Equation <em>y = f(x)<\/em><\/small><\/label>\n        <span id=\"inputField1\" contenteditable=\"true\" class=\"math-input\"><\/span>\n    <\/div>\n\n    <!-- \u0406\u043d\u0448\u0456 \u0435\u043b\u0435\u043c\u0435\u043d\u0442\u0438 \u043a\u0430\u043b\u044c\u043a\u0443\u043b\u044f\u0442\u043e\u0440\u0430 -->\n    <div class=\"parameter-container\">\n    <label for=\"lower_limit\">Initial Point <em>x<sub>0<\/sub><\/em><\/label>\n    <input type=\"number\" id=\"lower_limit\" name=\"lower_limit\" required \/>\n\n    <label for=\"upper_limit\">Final Point <em>x<sub>n<\/sub><\/em><\/label>\n    <input type=\"number\" id=\"upper_limit\" name=\"upper_limit\" required \/>\n\n    <label for=\"initial_condition\">Initial Value <em>y<sub>0<\/sub><\/em><\/label>\n    <input type=\"number\" id=\"initial_condition\" name=\"initial_condition\" required \/>\n\n    <label for=\"num_partitions\">Number of Steps (Partition of <em>[x<sub>0<\/sub>, x<sub>n<\/sub>]<\/em>)<\/label>\n    <input type=\"number\" id=\"num_partitions\" name=\"num_partitions\" required>\n\n    <label for=\"type\">Choose Euler's Method Variant<\/label>\n    <select id=\"type\" name=\"type\">\n        <option selected value=\"0\">Classic Euler's Method<\/option>\n        <option value=\"1\">Modified Euler's Method<\/option>\n        <option value=\"2\">Euler's-Cauchy Method<\/option>\n    <\/select>\n    \n    <\/div>\n\n    <!-- \u041a\u043d\u043e\u043f\u043a\u0430 \u0437\u0430\u043f\u0443\u0441\u043a\u0443 \u043e\u0431\u0447\u0438\u0441\u043b\u0435\u043d\u043d\u044f -->\n    <input type=\"button\" id=\"generate-button\" name=\"integral\" value=\"Calculate\" onclick=\"calculateIntegral()\" \/>\n<\/div>\n<div id=\"calculation-results\" style=\"display: none;\">\n        \n    <div id=\"fcontainer\" class=\"solution-container\"><\/div>\n<\/div>\n<div id=\"error-block\" class=\"error-message\"><\/div>\n<div id=\"loading\" class=\"loader\" style=\"display: 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5px;\n        font-size: 16px;\n        cursor: pointer;\n        margin-top: 10px;\n        transition: background-color 0.3s ease;\n    }\n    \n    #generate-button:hover {\n        background-color: #065a71;\n    }\n    \n    \/* \u041a\u043e\u043d\u0442\u0435\u0439\u043d\u0435\u0440 \u0434\u043b\u044f \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u0456\u0432 *\/\n    .solution-container {\n        background-color: #ffffff;\n        padding: 20px;\n        margin-top: 15px;\n        margin-bottom: 15px;\n        font-size: 16px;\n        color: #333;\n        line-height: 1.6;\n        max-width: 800px;\n        margin-left: auto;\n        margin-right: auto;\n        overflow-x: auto;\n    }\n    \n    \/* \u0410\u043d\u0456\u043c\u0430\u0446\u0456\u044f \u0437\u0430\u0432\u0430\u043d\u0442\u0430\u0436\u0435\u043d\u043d\u044f *\/\n    .loader {\n        display: block;\n        margin: 20px auto;\n        border: 8px solid #f3f3f3;\n        border-top: 8px solid 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\u043f\u043e\u043b\u044f\n    let answerMathField1 = MQ.MathField(inputSpan1, {\n      spaceBehavesLikeTab: true,\n      handlers: {\n        edit: function() {\n          activeMathField = answerMathField1; \/\/ \u0412\u0441\u0442\u0430\u043d\u043e\u0432\u043b\u044e\u0454\u043c\u043e \u0430\u043a\u0442\u0438\u0432\u043d\u0435 \u043f\u043e\u043b\u0435\n        }\n      }\n    });\n    \n    \/\/ **\u041e\u043d\u043e\u0432\u043b\u044e\u0454\u043c\u043e activeMathField \u043f\u0440\u0438 \u043d\u0430\u0442\u0438\u0441\u043a\u0430\u043d\u043d\u0456 \u043d\u0430 \u043f\u043e\u043b\u0435**\n    inputSpan.addEventListener('click', () => {\n      activeMathField = answerMathField;\n    });\n    \n    inputSpan1.addEventListener('click', () => {\n      activeMathField = answerMathField1;\n    });\n    \n    \/\/ **\u041e\u043d\u043e\u0432\u043b\u044e\u0454\u043c\u043e activeMathField \u043f\u0440\u0438 \u0444\u043e\u043a\u0443\u0441\u0443\u0432\u0430\u043d\u043d\u0456**\n    inputSpan.addEventListener('focus', () => {\n      activeMathField = answerMathField;\n      answerMathField.focus();\n    });\n    \n    inputSpan1.addEventListener('focus', () => {\n      activeMathField = answerMathField1;\n      answerMathField1.focus();\n    });\n    \n    \/\/ **\u0424\u0443\u043d\u043a\u0446\u0456\u044f \u0434\u043b\u044f \u0434\u043e\u0434\u0430\u0432\u0430\u043d\u043d\u044f \u0441\u0438\u043c\u0432\u043e\u043b\u0456\u0432 \u0443 \u0430\u043a\u0442\u0438\u0432\u043d\u0435 \u043f\u043e\u043b\u0435**\n    function addSymbol(symbol) {\n      if (!activeMathField) {\n        activeMathField = answerMathField; \/\/ \u0417\u0430 \u0437\u0430\u043c\u043e\u0432\u0447\u0443\u0432\u0430\u043d\u043d\u044f\u043c \u043f\u0435\u0440\u0448\u0435 \u043f\u043e\u043b\u0435\n      }\n      activeMathField.write(symbol);\n      activeMathField.focus();\n    }\n    \n    \/\/ **\u041e\u0431\u0440\u043e\u0431\u043a\u0430 \u0432\u0432\u0435\u0434\u0435\u043d\u043d\u044f 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methodType = document.getElementById('type');\n\n        \/\/ \u0406\u043d\u0456\u0446\u0456\u0430\u043b\u0456\u0437\u0430\u0446\u0456\u044f \u0434\u0432\u043e\u0432\u0438\u043c\u0456\u0440\u043d\u043e\u0433\u043e \u043c\u0430\u0441\u0438\u0432\u0443 \u0434\u043b\u044f \u0437\u0431\u0435\u0440\u0435\u0436\u0435\u043d\u043d\u044f \u0432\u0432\u0435\u0434\u0435\u043d\u0438\u0445 \u043a\u043e\u0440\u0438\u0441\u0442\u0443\u0432\u0430\u0447\u0435\u043c \u0434\u0430\u043d\u0438\u0445\n        var enteredData = [];\n        \/\/ \u0414\u043e\u0434\u0430\u0432\u0430\u043d\u043d\u044f \u0437\u043d\u0430\u0447\u0435\u043d\u043d\u044f \u0434\u043e \u0434\u0432\u043e\u0432\u0438\u043c\u0456\u0440\u043d\u043e\u0433\u043e \u043c\u0430\u0441\u0438\u0432\u0443\n        enteredData.push(func || '');\n        enteredData.push(func1 || '');\n        enteredData.push(parseFloat(lowLimit.value) || 0);\n        enteredData.push(parseFloat(uppLimit.value) || 0);\n        enteredData.push(parseFloat(initialCondition.value) || 0);\n        enteredData.push(parseFloat(numPartitions.value) || 0);\n        enteredData.push(parseFloat(methodType.value) || 0);\n        \n        sendDataToServer(enteredData);\n        \n        transferX = document.getElementById(\"X\");\n        transferY = document.getElementById(\"Y\");\n        size = document.getElementById(\"SIZE\");\n        transferX.value = 0;\n        transferY.value = 0;\n        size.value = 0;\n    }\n    \n\n    function drawGraphAndLines(graphData, linesData) {\n        const svgWidth = 600;\n        const svgHeight = 350;\n        const margin = { top: 20, right: 20, bottom: 30, left: 50 };\n    \n        const width = svgWidth - margin.left - margin.right;\n        const height = svgHeight - margin.top - margin.bottom;\n    \n        let formattedGraphData = convertDataForSVG(graphData);\n        let formattedLinesData = convertDataForSVG(linesData);\n    \n        const xMin = d3.min(formattedGraphData, d => d.x);\n        const xMax = d3.max(formattedGraphData, d => d.x);\n        let yMin = d3.min(formattedGraphData, d => d.y);\n        yMin = yMin > 0 ? 0 : yMin;\n        let yMax = d3.max(formattedGraphData, d => d.y);\n        yMax = yMax < 0 ? 0 : yMax;\n    \n        \/\/ Clear and set up the SVG element\n        \/*const svg = d3.select(\"#graph-svg\")\n            .html(\"\") \/\/ Clear previous contents\n            .attr(\"width\", svgWidth)\n            .attr(\"height\", svgHeight)\n            .append(\"g\")\n            .attr(\"transform\", `translate(${margin.left},${margin.top})`);*\/\n            \n        const svg = d3.select(\"#graph-svg\")\n            .html(\"\") \/\/ \u041e\u0447\u0438\u0449\u0430\u0454\u043c\u043e \u043f\u043e\u043f\u0435\u0440\u0435\u0434\u043d\u0456\u0439 \u0433\u0440\u0430\u0444\u0456\u043a\n            .attr(\"viewBox\", `0 0 ${svgWidth} ${svgHeight}`) \/\/ \u0410\u0434\u0430\u043f\u0442\u0438\u0432\u043d\u0435 \u043c\u0430\u0441\u0448\u0442\u0430\u0431\u0443\u0432\u0430\u043d\u043d\u044f\n            .attr(\"preserveAspectRatio\", \"xMidYMid meet\") \/\/ \u0417\u0431\u0435\u0440\u0435\u0436\u0435\u043d\u043d\u044f \u043f\u0440\u043e\u043f\u043e\u0440\u0446\u0456\u0439\n            .attr(\"width\", \"100%\")  \/\/ \u0410\u0434\u0430\u043f\u0442\u0438\u0432\u043d\u0430 \u0448\u0438\u0440\u0438\u043d\u0430\n            .attr(\"height\", \"100%\") \/\/ \u0410\u0434\u0430\u043f\u0442\u0438\u0432\u043d\u0430 \u0432\u0438\u0441\u043e\u0442\u0430\n            .append(\"g\")\n            .attr(\"transform\", `translate(${margin.left},${margin.top})`);\n    \n        const xScale = d3.scaleLinear()\n            .domain([xMin, xMax])\n            .range([0, width]);\n    \n        const yScale = d3.scaleLinear()\n            .domain([yMin, yMax])\n            .range([height, 0]);\n  \n\n        \/\/ Append the axes\n        svg.append(\"g\")\n            .attr(\"transform\", `translate(0,${height})`)\n            .call(d3.axisBottom(xScale))\n            .selectAll(\"path, line\")\n            .attr(\"stroke\", \"#E8E8E8\");\n    \n        svg.append(\"g\")\n            .call(d3.axisLeft(yScale))\n            .selectAll(\"path, line\")\n            .attr(\"stroke\", \"#E8E8E8\");\n    \n        \/\/ Add grid lines\n        svg.append(\"g\")\n            .attr(\"class\", \"grid\")\n            .call(\n                d3.axisLeft(yScale)\n                    .tickSize(-width)\n                    .tickFormat(\"\")\n                    .ticks(10)\n            )\n            .selectAll(\"line\")\n            .attr(\"stroke\", \"#E8E8E8\")\n            .attr(\"stroke-opacity\", 1)\n            .filter((d, i, nodes) => d !== yMax)\n            .attr(\"stroke-dasharray\", \"\");\n    \n        svg.selectAll(\".domain\")\n            .attr(\"stroke\", \"#E8E8E8\");\n    \n        \/\/ Define the area between the graph and the OX axis\n        const area = d3.area()\n            .x(d => xScale(d.x))\n            .y0(yScale(0)) \/\/ OX at y=0\n            .y1(d => yScale(d.y));\n    \n        \/\/ Define the line generator for the function graph\n        const line = d3.line()\n            .x(d => xScale(d.x))\n            .y(d => yScale(d.y));\n    \n        \/\/ Append the function graph line\n        svg.append(\"path\")\n            .datum(formattedGraphData)\n            .attr(\"fill\", \"none\")\n            .attr(\"stroke\", \"#0b638b\")\n            .attr(\"stroke-width\", 3)\n            .attr(\"d\", line);\n    \n        \/\/ Draw the Euler line (the broken line) using linesData\n        const eulerLine = d3.line()\n            .x(d => xScale(d.x))\n            .y(d => yScale(d.y));\n    \n        svg.append(\"path\")\n            .datum(formattedLinesData)\n            .attr(\"fill\", \"none\")\n            .attr(\"stroke\", \"#FF0000\")         \/\/ Red color\n            .attr(\"stroke-width\", 2)\n            .attr(\"d\", eulerLine);\n    \n        \/\/ ---------------------------\n        \/\/ Draw division points\n        \/\/ ---------------------------\n    \n        \/\/ (1) Division points on the Euler line (using linesData)\n        svg.selectAll(\".euler-division-point\")\n            .data(formattedLinesData)\n            .enter()\n            .append(\"circle\")\n            .attr(\"class\", \"euler-division-point\")\n            .attr(\"cx\", d => xScale(d.x))\n            .attr(\"cy\", d => yScale(d.y))\n            .attr(\"r\", 4)\n            .attr(\"fill\", \"#FF0000\");\n    \n        \/\/ (2) Division points on the function graph.\n        \/\/ For each division point from linesData, find the closest point on the function graph.\n        const divisionPointsFunction = formattedLinesData.map(p => {\n            let closest = formattedGraphData.reduce((prev, curr) => {\n                return (Math.abs(curr.x - p.x) < Math.abs(prev.x - p.x)) ? curr : prev;\n            });\n            return { x: p.x, y: closest.y };\n        });\n    \n        svg.selectAll(\".graph-division-point\")\n            .data(divisionPointsFunction)\n            .enter()\n            .append(\"circle\")\n            .attr(\"class\", \"graph-division-point\")\n            .attr(\"cx\", d => xScale(d.x))\n            .attr(\"cy\", d => yScale(d.y))\n            .attr(\"r\", 4)\n            .attr(\"fill\", \"#0b638b\");\n    \n        \/\/ ---------------------------\n        \/\/ Add vertical dotted line and point display functionality\n        \/\/ ---------------------------\n        const focusLine = svg.append(\"line\")\n            .attr(\"stroke\", \"gray\")\n            .attr(\"stroke-dasharray\", \"3,3\")\n            .attr(\"y1\", 0)\n            .attr(\"y2\", height)\n            .style(\"display\", \"none\");\n    \n        const focusCircle = svg.append(\"circle\")\n            .attr(\"r\", 4)\n            .attr(\"fill\", \"#0b638b\")\n            .style(\"display\", \"none\");\n    \n        const focusText = svg.append(\"text\")\n            .attr(\"fill\", \"#333\")\n            .style(\"display\", \"none\");\n    \n        svg.append(\"rect\")\n            .attr(\"width\", width)\n            .attr(\"height\", height)\n            .attr(\"fill\", \"none\")\n            .attr(\"pointer-events\", \"all\")\n            .on(\"mousemove\", mousemove)\n            .on(\"mouseout\", () => {\n                focusLine.style(\"display\", \"none\");\n                focusCircle.style(\"display\", \"none\");\n                focusText.style(\"display\", \"none\");\n            });\n    \n        function mousemove(event) {\n            const [mouseX] = d3.pointer(event);\n            const xValue = xScale.invert(mouseX);\n    \n            if (xValue < xMin || xValue > xMax) {\n                focusLine.style(\"display\", \"none\");\n                focusCircle.style(\"display\", \"none\");\n                focusText.style(\"display\", \"none\");\n                return;\n            }\n    \n            const closestPoint = formattedGraphData.reduce((prev, curr) =>\n                Math.abs(curr.x - xValue) < Math.abs(prev.x - xValue) ? curr : prev\n            );\n    \n            const xPos = xScale(closestPoint.x);\n            const yPos = yScale(closestPoint.y);\n    \n            if (yPos < 0 || yPos > height) {\n                focusLine.style(\"display\", \"none\");\n                focusCircle.style(\"display\", \"none\");\n                focusText.style(\"display\", \"none\");\n                return;\n            }\n    \n            focusLine\n                .attr(\"x1\", xPos)\n                .attr(\"x2\", xPos)\n                .attr(\"stroke-dasharray\", \"1,3\")\n                .style(\"display\", null);\n    \n            focusCircle\n                .attr(\"cx\", xPos)\n                .attr(\"cy\", yPos)\n                .style(\"display\", null);\n    \n            const textPadding = 10;\n            let textX = xPos + textPadding;\n            let textY = yPos - textPadding;\n    \n            if (xPos + 100 > width) {\n                textX = xPos - 100;\n            }\n    \n            if (yPos - 20 < 0) {\n                textY = yPos + textPadding;\n            }\n    \n            focusText\n                .attr(\"x\", textX)\n                .attr(\"y\", textY)\n                .text(`x: ${closestPoint.x.toFixed(2)}, y: ${closestPoint.y.toFixed(2)}`)\n                .attr(\"font-size\", 10)\n                .style(\"display\", null);\n        }\n    }\n\n    \/\/ \u0414\u043e\u043f\u043e\u043c\u0456\u0436\u043d\u0430 \u0444\u0443\u043d\u043a\u0446\u0456\u044f \u0434\u043b\u044f \u043a\u043e\u043d\u0432\u0435\u0440\u0442\u0430\u0446\u0456\u0457 \u0434\u0430\u043d\u0438\u0445, \u043e\u0442\u0440\u0438\u043c\u0430\u043d\u0438\u0445 \u0432\u0456\u0434 \u0441\u0435\u0440\u0432\u0435\u0440\u0430, \u0443 \u0444\u043e\u0440\u043c\u0430\u0442 {x, y}\n    function convertDataForSVG(graphData) {\n        return graphData.map(point => ({ x: point[0], y: point[1] }));\n    }\n\n\/\/-------------------------------------------------------------\n    \n    function renderMathJax() {\n        MathJax.Hub.Queue([\"Typeset\", MathJax.Hub]);\n    }\n    \n    function sendDataToServer(enteredData) {\n        var loading = document.getElementById('loading');\n        var resultC = document.getElementById('fcontainer');\n        var resultCalc = document.getElementById('calculation-results');\n        var errorBlock = document.getElementById('error-block'); \/\/ \u0411\u043b\u043e\u043a \u0434\u043b\u044f \u043f\u043e\u0432\u0456\u0434\u043e\u043c\u043b\u0435\u043d\u044c \u043f\u0440\u043e \u043f\u043e\u043c\u0438\u043b\u043a\u0438\n        \n        resultC.innerHTML = '';\n        errorBlock.classList.remove('show'); \/\/ \u041f\u0440\u0438\u0445\u043e\u0432\u0443\u0454\u043c\u043e \u0431\u043b\u043e\u043a \u043f\u043e\u043c\u0438\u043b\u043e\u043a \u043f\u0435\u0440\u0435\u0434 \u043f\u043e\u0447\u0430\u0442\u043a\u043e\u043c \u043d\u043e\u0432\u043e\u0433\u043e \u0437\u0430\u043f\u0438\u0442\u0443\n        errorBlock.innerHTML = ''; \n    \n        loading.style.display = 'block';\n        resultCalc.style.display = 'none';\n    \n        var formData = new FormData();\n        formData.append('action', 'differentiation_by_eulers_method');\n        formData.append('enteredData', JSON.stringify(enteredData));\n\n        fetch('\/en\/wp-admin\/admin-ajax.php', {\n            method: 'POST',\n            body: formData\n        })\n        .then(response => {\n           if (!response.ok) {\n                throw new Error('Network response was not ok');\n            }\n            return response.json();\n        })\n        .then(data => {\n            loading.style.display = 'none';\n            if (data.success) {\n                if (Array.isArray(data.result)) {\n                    const messages = [\n                        \"Your Results Are Ready! Here Are the Calculated Values\",\n                        \"Here\u2019s Your Result! Check Out the Calculations\",\n                        \"Calculation Complete! Take a Look at Your Values\",\n                        \"Your Data Has Been Processed! Here\u2019s the Answer\",\n                        \"Computation Finished! 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You simply enter the equation in an easy-to-use format, specify the initial conditions, and choose the number of intermediate points. The calculator automatically divides the given interval into equal parts and computes function values step by step, applying the <a title=\"How Euler\u2019s Method Works\" href=\"https:\/\/www.mathros.net.ua\/en\/eulers-method.html\">classic Euler\u2019s method<\/a> or its <a title=\"Modified Euler\u2019s Method\" href=\"https:\/\/www.mathros.net.ua\/en\/modified-eulers-method.html\">improved variations<\/a>.<\/p>\n<p class=\"calculator-p\">Once the calculations are complete, you\u2019ll get a <strong>clear step-by-step table<\/strong> that shows the argument values, approximate function values, and exact results for comparison. Additionally, the calculator <strong>visualizes the solution as a graph<\/strong>, making it easier to understand how the function behaves. It also calculates the <strong>approximation error<\/strong>, helping you evaluate the accuracy of the method.<\/p>\n<p class=\"calculator-p\">For more flexibility, you can choose between the <strong>modified Euler's method<\/strong> or the <strong>Euler's-Cauchy method<\/strong>. These enhanced versions provide higher accuracy by incorporating additional derivative estimates, not just the function\u2019s current value. This allows you to <strong>compare different approaches<\/strong> and select the most effective method for your specific problem.<\/p>\n<p class=\"calculator-p\">With the <strong>Euler's method calculator<\/strong>, solving differential equations becomes quick, intuitive, and insightful\u2014giving you a deeper understanding of numerical solutions with just a few clicks!<\/p>\n<style>\n    .h2-title2 {\n        text-align: center; \/* \u0426\u0435\u043d\u0442\u0440\u0443\u0432\u0430\u043d\u043d\u044f \u0437\u0430\u0433\u043e\u043b\u043e\u0432\u043a\u0430 *\/\n        color: gray; \/* \u0417\u0431\u0435\u0440\u0435\u0436\u0435\u043d\u043d\u044f \u0441\u0456\u0440\u043e\u0433\u043e \u043a\u043e\u043b\u044c\u043e\u0440\u0443 \u0442\u0435\u043a\u0441\u0442\u0443 *\/\n        font-size: 28px; \/* \u0417\u0430\u043b\u0438\u0448\u0430\u0454\u043c\u043e \u0440\u043e\u0437\u043c\u0456\u0440 \u0442\u0435\u043a\u0441\u0442\u0443 *\/\n        font-weight: 700; \/* \u0420\u043e\u0431\u0438\u043c\u043e \u0442\u0435\u043a\u0441\u0442 \u0436\u0438\u0440\u043d\u0438\u043c \u0434\u043b\u044f \u0430\u043a\u0446\u0435\u043d\u0442\u0443 *\/\n        line-height: 1.4; 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