How to Solve a First-Order Differential Equation in Excel (Easy Steps)

Understand Differential Equations

  • A differential equation relates one or more functions to their derivatives with respect to an independent variable.
  • The highest order of the derivative determines the order of the differential equation.
  • Differential equations are used to model growth, decay, and various phenomena, including physics applications.

Problem Statement

  • Given the differential equation:

    dy/dx=f(x)
  • Initial condition:

  • We want to compute the values of

    in the time interval

    .


STEP 1 – Create Dataset

  • Create a dataset to insert the iteration number (n), time value (t), y value, yexact value, and delta_t.
  • The delta_t is the separation value.
  • Display the equations to find the t(n+1) and y(n+1) values.
  • Add the time interval and initial value to make the calculation easy.

Step-by-Step Procedures to Solve Differential Equation in Excel


STEP 2 – Apply Formula for Time Values & Y-values

  • Apply formulas to calculate the time values and y-values.
  • At the start of the iteration, n is 0, t is 0, and y is 5.
  • Insert these values in Cells B5, C5, and D5 respectively.

Step-by-Step Procedures to Solve Differential Equation in Excel

  • Insert a formula to calculate the t-values.
  • Select Cell C6 and enter the formula below:
=C5+$G$5
  • Press Enter.

Step-by-Step Procedures to Solve Differential Equation in Excel

  • To find the y-values:
    • Select Cell D6 and enter the formula below:
=D5+(C5-D5)*$G$5
    • Press Enter to see the result.

Step-by-Step Procedures to Solve Differential Equation in Excel


STEP 3 – Fill Columns

  • To fill the values in the n, t, and columns:
    • To find the number of iterations (n), divide the time interval by the delta_t value.
    • In this problem statement the iteration number (n) = (2/0.1) = 12.
    • Fill the iteration column from 0 to 12.

Step-by-Step Procedures to Solve Differential Equation in Excel

    • Select both Cells C6 & D6 and drag the Fill Handle down.

Step-by-Step Procedures to Solve Differential Equation in Excel

    • The t and yvalues have been calculated.

Step-by-Step Procedures to Solve Differential Equation in Excel


STEP 4 – Plot Data

  • To plot the t and yvalues, select the range C4:D17.

Step-by-Step Procedures to Solve Differential Equation in Excel

  • Go to the Insert tab and click on the Insert Scatter icon. A drop-down menu will appear.
  • Select Scatter with Lines.

Step-by-Step Procedures to Solve Differential Equation in Excel

  • As a result, a chart will display on the sheet.
  • Change the chart title according to your preferences.

Step-by-Step Procedures to Solve Differential Equation in Excel


STEP 5 – Format Axis

  • To format the axis of the graph:
    • Doubleclick on the horizontal axis. It will open the Format Axis options on the right side of the sheet.
    • Set the Minimum value to 0 and Maximum value to 1.2 in the Bounds section.
    • Press Enter.

Step-by-Step Procedures to Solve Differential Equation in Excel

    • Doubleclick on the vertical axis.
    • Set the Minimum value to 0.3 and Maximum value to 0.65 in the Bounds section.
    • Press Enter.

Step-by-Step Procedures to Solve Differential Equation in Excel

  • The graph will look like the picture below:

Step-by-Step Procedures to Solve Differential Equation in Excel

Read More: How to Solve an Equation for X When Y is Given in Excel 


STEP 6 – Find the Exact Solution

  • The exact solution of the differential equation is y = -1 + t + 1.5*exp(-t).
  • Select Cell E5 and enter the formula below:
=-1+C5+1.5*EXP(-C5)

Step-by-Step Procedures to Solve Differential Equation in Excel

  • Press Enter and drag the Fill Handle down.

Step-by-Step Procedures to Solve Differential Equation in Excel

  • The y-values of the exact solution have been calculated.


STEP 7 – Plot the Exact Solution on the Same Chart

  • To compare the y-values of the exact solution with the numerical values, plot them on the same chart.
  • Rightclick on the chart to open the menu.
  • Click on the Select Data option.

  • In the Select Data Source dialog box, click on Add.

  • Insert the Series X and Y values in the Edit Series box.

  • Click OK to get the plot of the exact solution.

Read More: How to Solve 2 Equations with 2 Unknowns in Excel


Final Output

  • If the delta_t value in Cell G5 is changed, the whole dataset will automatically update and show the solution for the updated delta_t value.

  • The graphs will also update automatically.

Read More: How to Solve Polynomial Equation in Excel


Download Practice Book

You can download the practice workbook from here:


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Mursalin Ibne Salehin
Mursalin Ibne Salehin

Mursalin Ibne Salehin holds a BSc in Electrical and Electronics Engineering from Bangladesh University of Engineering and Technology. Over the past 2 years, he has actively contributed to the ExcelDemy project, where he authored over 150 articles. He has also led a team with content development works. Currently, he is working as a Reviewer in the ExcelDemy Project. He likes using and learning about Microsoft Office, especially Excel. He is interested in data analysis with Excel, machine learning,... Read Full Bio

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