Verifying Solutions to Differential Equations, Examples - Calculus

To verify a solution to a differential equation, take the derivatives of the proposed solution as needed to match the order of the differential equation. Then, substitute the original function and its derivatives into the original differential equation. If the left side of the equation equals the right side, then the proposed function is a valid solution.

💡Steps to Verify a Solution
• Identify the proposed solution and the differential equation.
• Calculate the necessary derivatives: of the proposed solution. For example, if the differential equation involves a first derivative, calculate the first derivative of the function; if it involves a second derivative, calculate both the first and second derivatives.
• Substitute the function and its derivatives: into the original differential equation.
• Simplify both sides of the equation .
• Check for consistency . If the left-hand side (LHS) of the equation is equal to the right-hand side (RHS), the function is a solution. If LHS does not equal RHS, it is not a solution.

💡Example
Consider the differential equation dy/dx = 3x^2 and the proposed solution y = x^3.
• Identify:
⚬ Differential equation: dy/dx = 3x^2
⚬ Proposed solution: y = x^3
• Calculate derivatives:
⚬ dy/dx = 3x^2 (using the power rule).
• Substitute:
⚬ Replace dy/dx in the differential equation with the calculated derivative: 3x^2 = 3x^2.
• Simplify:
⚬ The equation is already in its simplest form.
• Check for consistency:
⚬ 3x^2 = 3x^2 is true for all values of x, so y = x^3 is a verified solution.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1DMK4EA0f8SfF4SgdiOZ39F73He_YWIwe/view?usp=drive_link
• Answers: https://drive.google.com/file/d/1QpzjiCPjfoxydzZqvjyJJsqv9jyRLZ0o/view?usp=drive_link

💡Chapters:
00:00 Verifying solutions to ODEs
01:18 Worked example

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