Question

Consider the differential equation x^2y′′ − 3xy′ − 5y = 0. Note that this is not a constant coefficient differential equation, but it is linear. The theory of linear differential equations states that the dimension of the space of all homogeneous solutions equals the order of the differential equation, so that a fundamental solution set for this equation should have two linearly fundamental solutions.

• Assume that y = x^r is a solution. Find the resulting characteristic equation for r. • Write the general solution of the differential equation.
• Find the solution for y(1) = 2 and y′(1) = 1.

Answer 1

We get

Thus, the characteristic equation is

Since , the solutions to this equation are . Thus, the general solution is

Note that . If and then

Solving the equations

we get and . Hence, the solution is