91Ó°ÊÓ

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.