Question

Solve the following initial value problem. x 2 y ′′+13xy ′ +136y=0,y(1)=1,y ′

(1)=9 Enter your answer as a symbolic function of x, as in these Do not include ' y= ' in your answer. examples

Answer 1

To solve the given initial value problem, we first find the auxiliary equation by substituting y = e^(rx) into the differential equation. Then, we determine the general solutions for each region and use the initial conditions to find the particular solution.

Step-by-step explanation:

To solve the given initial value problem, we first need to find the auxiliary equation by substituting y = e^(rx) into the differential equation. This gives us r^2x^2 + 13rx + 136 = 0. Solving this quadratic equation, we find that r = -4 and r = -8.

Therefore, the general solution for region I is yI(x) = c1x^(-4) + c2x^(-8), and the general solution for region III is yIII(x) = c3e^(-8x) + c4e^(-4x).

Now, we can determine the particular solution by using the initial conditions. Plugging in x = 1 and y = 1 into yIII(x) = c3e^(-8x) + c4e^(-4x), we get 1 = c3e^(-8) + c4e^(-4) and y' = -8c3e^(-8) - 4c4e^(-4).