Question

Find the solution to the initial value problem:

y''(θ) - y(θ) = 5sin(θ) - 3e²θ
with initial conditions: y(0) = 1 and y'(0) = -1.
Determine the function y(θ) that satisfies this differential equation and these initial conditions.

Answer 1

To find the function y(θ) that satisfies the initial value problem and conditions given, one must solve the homogeneous differential equation, find a particular solution for the nonhomogeneous part, and apply the initial conditions to determine the constants.

Step-by-step explanation:

The solution to the initial value problem y''(θ) - y(θ) = 5sin(θ) - 3e²θ with initial conditions y(0) = 1 and y'(0) = -1 is found by solving the homogeneous equation y''(θ) - y(θ) = 0 and finding a particular solution to the nonhomogeneous equation.

The general solution of the homogeneous equation is yh(θ) = C1eθ + C2e-θ where C1 and C2 are constants determined by the initial conditions.

To find a particular solution yp(θ) to the nonhomogeneous equation, we assume yp(θ) = Asin(θ) + Be2θ and substitute into the differential equation to determine A and B. The complete solution is the sum y(θ) = yh(θ) + yp(θ).

Applying the initial conditions to the complete solution gives us two equations to solve for C1 and C2. After which, we combine them to find the specific function y(θ) that satisfies the entire initial value problem.