Question

Solve the given initial-value problem. xy' + y = ex, y(1) = 5 Give the largest interval I over which the solution is defined. (Enter your answer using interval notation)

Answer 1

The initial-value problem xy' + y = e^x, y(1) = 5, is a first-order linear ODE that can be solved using integrating factors. The solution is y = (e^x + 5 - e)/|x|, and the solution is defined over the interval I = (-∞, 0) ∪ (0, ∞).

Step-by-step explanation:

To solve the initial-value problem xy' + y = ex, y(1) = 5, we need to recognize it as a first-order linear ordinary differential equation that can be resolved using an integrating factor. This ODE can be written in the form y' + P(x)y = Q(x), where P(x) = 1/x and Q(x) = ex/x.

First, find the integrating factor μ(x) which is e∫ P(x)dx = eln|x| = |x|. Then multiply through by this integrating factor to get |x|y' + |x|(1/x)y = ex. Now, because we have a product |x|y' + y, it suggests the left-hand side is the derivative of |x|y. So the equation simplifies to (|x|y)' = ex, which we integrate both sides with respect to x to get |x|y = ∫ exdx = ex + C.

Applying the initial condition y(1) = 5, we find that 1*5 = e1 + C, which gives us C = 5 - e. The solution to the ODE is y = (ex + 5 - e)/|x|. The largest interval I in which the solution is defined is all real numbers except for x = 0; hence, I = (-∞, 0) ∪ (0, ∞).