Question

What is the difference between a solution of a differential equation and a solution of an initial-value problem?

Short answer

A solution of a differential equation has many solutions whereas the solution of a initial value problem has one and only solution that satisfies the initial condition of the equation.

Step-by-step solution

Step 1. Explanation

A differential equation has many solutions which are functions that makes the differential equation true.

let us take the differential equation $\frac{dy}{dx}=x^2$,

Now, the solution of the above equation would be the functions whose derivatives equal $x^2$.

for example, the function role="math" localid="1649906001058" $y\left(x\right)=4x^3+C$would be the solution of the equation $\frac{dy}{dx}=x^2$.

A initial value problem has one and only solution which would satisfy the initial condition of the equation.

For example, In the differential equation $\frac{dy}{dx}=x^2$with initial condition $y\left(0\right)=5$, only the function of the form $y\left(x\right)=4x^3+5$which satisfies both as derivative of the equation and also satisfies the initial condition is the solution of the initial value problem.