Solve the given initial-value problem.

To solve the initial-value problem given by the differential equation with initial conditions and , we begin by rewriting the equation in standard form:

Next, we find the characteristic equation by substituting :

Using the quadratic formula :

This gives us the roots:

Thus, the general solution of the differential equation is:

We now apply the initial conditions to determine and .

Using the first initial condition :

Next, we compute :

Using the second initial condition :

Now we have the system of equations:

From (1), we can express in terms of :

Substituting into (2):

Multiplying everything by 2 to eliminate the fraction:

which simplifies to:

Now substituting back into (1):

So we have determined the constants:

Finally, substituting back into the general solution gives the particular solution:

Thus, the solution to the initial-value problem is: