7.2 Verifying Solutions for Differential Equations

Learning objectives

—FUN-7.B—Verify solutions to differential equations• FUN-7.B Verify solutions to differential equations.• FUN-7.B.1 Derivatives can be used to verify that a function is a solution to a given differential equation.• FUN-7.B.2 There may be infinitely many solutions to a differential equation.• Enduring understanding FUN-7: Solving differential equations allows us to determine functions and develop models.

To verify a candidate solution, compute every derivative named in the differential equation and substitute both the candidate function and those derivatives. It is a solution on an interval only if the left- and right-hand sides simplify to the same expression for every input in that interval.

Step	Verification move
1	identify the candidate function and relevant interval
2	compute the derivative or derivatives required
3	substitute into both sides of the differential equation
4	simplify and confirm an identity, not a one-point match
5	separately check any initial or boundary condition

Verify $y=3e^{2x}$ for $dy/dx=2y$ . Differentiation gives $dy/dx=6e^{2x}$ , while substituting the candidate on the right gives $2y=2(3e^{2x})=6e^{2x}$ . Both sides agree for every real $x$ , so the candidate is a solution.

More generally, $y=Ce^{2x}$ gives $dy/dx=2Ce^{2x}=2y$ for every constant $C$ . Thus the same differential equation has infinitely many solutions. An additional condition such as $y(0)=3$ selects $C=3$ from that family.

Matching the equation at a single point does not verify a solution, and satisfying an initial condition alone is not enough. The candidate must satisfy the differential equation throughout its stated domain and must also satisfy every extra condition provided.

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—7.3 Sketching Slope FieldsRead