7.2 Verifying Solutions for Differential Equations

What it means to verify a solution

A function is a solution to a differential equation if, when you substitute it (and its derivatives) into the equation, both sides are equal for all values of the domain. To verify, find the necessary derivatives of the proposed solution and plug everything into the original equation to check the equality holds true.

Step-by-step verification process

1) Differentiate the candidate function to get dy/dx (and higher derivatives if needed). 2) Substitute y and dy/dx into the differential equation. 3) Simplify both sides algebraically. If the equation becomes a true statement (like 0 = 0), the function is a valid solution. If not, it isn't.

Initial conditions and particular solutions

A general solution contains an arbitrary constant (like +C); a particular solution satisfies a specific initial condition, such as y(0) = 3. To verify a particular solution, confirm it satisfies the differential equation AND plug in the initial x-value to check it produces the correct y-value.

Key terms

Verifying a solution

Substituting a function and its derivative(s) into a differential equation to check if both sides are equal.

General solution

A solution to a differential equation that includes an arbitrary constant, representing a family of functions.

Particular solution

A specific solution that satisfies both the differential equation and a given initial condition.

Initial condition

A given value, like y(x₀) = y₀, used to find a particular solution from a general solution.

Steps to verify a solution

1) Differentiate the candidate function. 2) Substitute y and derivatives into the equation. 3) Simplify and check for a true equality.

Try an AP-style question

Which function is a solution to y' = 2y?

1. y = x² + 5
2. y = 2eˣ
3. y = e^(x²)
4. y = 5e^(2x)