Question

To find the general solution to the homogeneous differential equation dx² /d²y+dx /dy−2y=0,

we can look for solutions of the form y=C1f1(x)+C2f2(x). To do this, let's first find the characteristic equation by assuming a solution of the form y=erx. Then we'll find f1(x) and f2(x).

Answer 1

The homogeneous differential equation is solved by finding a characteristic equation from assuming solutions of the form erx. Factoring and solving the characteristic equation yields the general solution with two exponential functions.

Step-by-step explanation:

To solve the homogenous differential equation dx²/d²y + dx/dy - 2y = 0, we consider solutions of the form y = erx. This leads us to the characteristic equation r² + r - 2 = 0, which can be solved to find the roots of the characteristic equation. These roots, say r1 and r2, will give us the functions f1(x) and f2(x) which form the general solution to the differential equation.

The roots are found by factoring the characteristic equation as (r+2)(r-1) = 0, yielding r1 = -2 and r2 = 1. Therefore, the general solution to our differential equation is y = C1e-2x + C2ex, where C1 and C2 are constants determined by initial or boundary conditions.