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

Understanding 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:

  $$y(0)=0.5$$

• We want to compute the values of

  $$y$$

  in the time interval

  $$[0,1.2]$$.

STEP 1 – Create a Dataset

• Create a dataset to insert the iteration number (n), time value (t), y value, y–exact 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 2 – Apply the 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.

• Insert a formula to calculate the t-values.
• Select Cell C6 and enter the formula below:

• Press Enter.

• To find the y-values:
  • Select Cell D6 and enter the formula below:

• • Press Enter to see the result.

STEP 3 – Fill the Columns

• To fill the values in the n, t, and y 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.

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

• • The t and y–values have been calculated.

STEP 4 – Plot the Data

• To plot the t and y–values, select the range C4:D17.

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

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

STEP 5 – Format Axis

• To format the axis of the graph:
  • Double–click 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.

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

• The graph will look like the picture below:

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:

• Press Enter and drag the Fill Handle down.

• 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.
• Right–click 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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