Final answer:
To solve the given initial-value problem, we first find the complementary solution by solving the homogeneous equation. The general solution will be the sum of the complementary and particular solutions.
Step-by-step explanation:
To solve the given initial-value problem:
y" + 4y' + 4y = (5x)e⁽⁻²ˣ⁾, y(0) = 3, y'(0) = 9
The first step is to find the complementary solution by solving the homogeneous equation y" + 4y' + 4y = 0. The characteristic equation is r² + 4r + 4 = 0, which gives us the double root r = -2. The complementary solution is therefore y_c(x) = (c₁ + c₂x)e⁽⁻²ˣ⁾ where c₁ and c₂ are constants.
Next, we find the particular solution using the method of undetermined coefficients. We assume a particular solution of the form y_p(x) = Ax⁻²e⁽⁻²ˣ⁾. Plugging this into the original equation gives us the equation -20Ae⁽⁻²ˣ⁾ + 10Axe⁽⁻²ˣ⁾ = 5x. Solving this, we find that A = -1/10. Therefore, the particular solution is y_p(x) = -1/10x⁻²e⁽⁻²ˣ⁾.
The general solution to the initial-value problem is the sum of the complementary and particular solutions: y(x) = (c₁ + c₂x)e⁽⁻²ˣ⁾ - (1/10)x⁻²e⁽⁻²ˣ⁾. Using the initial conditions y(0) = 3 and y'(0) = 9, we can solve for the constants c₁ and c₂. Finally, we have the specific solution to the initial-value problem.