Recipe

Solve an initial value problem

Use when: a differential equation is paired with a specific value y(x_0)=y_0, and a single particular solution, not the whole family, is wanted

Goal: the one member of the general solution's family that passes through the given point

Taught in Initial value problems and particular solutions · extends Solve a separable differential equation

1. 1

   Run the ordinary solving procedure (separation, or whichever technique applies) all the way to the general solution, before touching the initial condition at all.

   Example

   \dfrac{dy}{dx}=x^2y separates and integrates to the general solution y=Ce^{x^3/3}, exactly as in the recipe above.

   Why: substituting the initial condition before integrating throws away the very constant the condition is meant to determine.

2. 2

   Plug x=x_0 and y=y_0 into the general solution and solve the resulting equation for C.

   Example

   y(0)=5: 5=Ce^{0^3/3}=Ce^0=C, so C=5.

   Why: the general solution is already the full family, so substituting one known point on the curve pins down exactly which member it is.

3. 3

   Substitute the value of C found in Step 2 back into the general solution.

   Example

   \boxed{y = 5e^{x^3/3}}
   • the initial point lands somewhere the general solution's family is well-behaved (the ordinary case for the equations in this course) — C is determined uniquely, as above.
   • the initial point lands exactly where the separation step divided by zero (a rare exception) — check whether the discarded constant solution itself satisfies the initial condition instead.

   Why: this is the one specific curve, out of the entire family, that actually passes through the point the problem specified.

4. 4

   Differentiate the particular solution to confirm it satisfies the original differential equation, and evaluate it at x_0 to confirm it hits y_0.

   Example

   \dfrac{d}{dx}\left[5e^{x^3/3}\right]=5e^{x^3/3}\cdot x^2 = x^2y, matching the original equation. And y(0)=5e^0=5, matching the given condition.

   Why: this catches an arithmetic slip in solving for C before it's reported as the final answer.