Separable Equations: Answer to Example 2

Example: Solve the initial value problem

Answer: This is a separable equation. Indeed, we have

Before we get into integration we need to look for the constant solutions. These are the roots of the equation . Since this equation has no real roots, we conclude that no-constant solution exists. Therefore, we proceed with the separation of the two variables and integration. We have

,

which gives

Since

and

,

we get

The initial condition y(0)=1 gives

The particular solution to the initial value problem is

,

or in the explicit form

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