Final answer:
To solve the initial value problem 4x²y′′ - 20x(x+1)y′ + (50x+35)y = 2x⁹/²e²ˣ, y(1) = -1/12e²-e⁵-2, y′(1) = -3/8, we can use the method of undetermined coefficients to find the particular solution. Afterwards, the general solution can be obtained by adding the complementary solution and the particular solution together.
Step-by-step explanation:
To solve the initial value problem 4x²y′′ - 20x(x+1)y′ + (50x+35)y = 2x⁹/²e²ˣ, y(1) = -1/12e²-e⁵-2, y′(1) = -3/8, we can use the method of undetermined coefficients. First, find the complementary solution by solving the equation 4x²r² - 20xr + 50r = 0. The roots of this equation are r = 0 and r = 5/2. Therefore, the complementary solution is y_c(x) = C₁x⁰ + C₂x⁵/².
Next, find the particular solution. Assume a particular solution of the form y_p(x) = Ax⁹/²e²ˣ. Substitute this into the original equation and solve for A to find the particular solution.
Finally, the general solution is y(x) = y_c(x) + y_p(x). Substitute the initial conditions y(1) = -1/12e²-e⁵-2 and y′(1) = -3/8 into the general solution to find the values of C₁ and C₂. Hence, obtain the complete solution to the initial value problem.