Question

Solve the given initial-value problem. xy' + y = eˣ, y(1) = 6 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 with y(1) = 6 is solved by using the method of separating variables and integration. The solution is y = (6/e)e^x/x, and the largest interval over which the solution is defined, excluding the point where x=0, is I = (-∞, 0) ∪ (0, +∞).

Step-by-step explanation:

To solve the initial-value problem xy' + y = ex, where y(1) = 6, we will use the method of separating variables and integration. First, we can rewrite the differential equation in the differential form suitable for separation of variables:

dy/y = exdx/x

Now integrate both sides:

1. ∫ dy/y = ln|y| + C1
2. ∫ exdx/x = Ei(x) + C2

Given that y(1) = 6, we find the constant of integration C by plugging in the initial values:

ln|6| = Ei(1) + C

Now solve for the constant C and write the general solution for y:

y = Aex/x

Next, use the initial condition to solve for A:

6 = A * e1/1

A = 6/e

Thus, the solution to the initial-value problem is:

y = (6/e)ex/x

The largest interval I over which the solution is defined is for all x except where x=0 (since division by zero is not allowed).

I = (-∞, 0) ∪ (0, +∞)