# Find the general solution of the homogeneous differential equation. {eq}y'' - 7y' + 10 y = 24e^x {/eq}

## Question:

Find the general solution of the homogeneous differential equation.

{eq}y'' - 7y' + 10 y = 24e^x {/eq}

## Second-Order Linear Differential Equation:

A second-order linear differential equation with constant coefficients is of the form {eq}ay''+by'+cy=g(t) {/eq}, where {eq}a, b, c {/eq} are the constant coefficients.

In the homogeneous case, the right hand side is equal to 0, that is: {eq}ay''+by'+cy= 0 {/eq}.

A solution of such a differential equation is found by looking at the characteristic equation which is:

{eq}ar^2+br+c=0 {/eq} to be solved for r.

If {eq}r_1, r_2 {/eq} are the roots of the characteristic equation, then as long as these roots are different, the theory gives that the solution to the homogeneous differential equation will be of the form:

{eq}y_h = c_1\cdot e^{r_1 \cdot t}+c_2\cdot e^{r_2 \cdot t} {/eq}, where {eq}c_1, c_2 {/eq}

are constants to be found using the initial conditions.

If the differential equations has an inhomogeneous term (right hand side is non zero), then the solution will be of the form

{eq}y=y_h(t)+y_p(t) {/eq}

Where, the first term is a solution of the homogeneous differential equation and the second term is a particular solution that satisfies the inhomogeneous problem.

To find {eq}y_p(t) {/eq} for the particular case, as long as the inhomogeneous term is a combination of polynomials and exponentials, a guess can be made of a particular form that depends on the form of the inhomogeneous right hand side so that the full solution satisfies the full differential equation together with the initial conditions.