Final answer:
To solve the given initial-value problem xy' + y = eˣ, y(1) = 10, we can use the method of integrating factors.
Step-by-step explanation:
To solve the given initial-value problem xy' + y = eˣ, y(1) = 10, we can use the method of integrating factors.
First, divide both sides of the equation by x to get the equation in standard form: y' + (1/x)y = eˣ/x.
Next, we identify the integrating factor, which is e^(∫1/x dx) = e^(ln|x|) = |x|. Multiplying the entire equation by the integrating factor, we get |x|y' + y = eˣ.
Now, we rewrite the left side of the equation as the derivative of the product of |x| and y: d/dx (|x|y) = eˣ.
Integrating both sides with respect to x, we have ∫d/dx (|x|y) dx = ∫eˣ dx. The integral of d/dx (|x|y) is simply |x|y, and the integral of eˣ is eˣ.
Therefore, |x|y = eˣ + C, where C is the constant of integration.
Substituting x = 1, y = 10 into the equation, we solve for C: |1| * 10 = e¹ + C, 10 = e + C. Subtracting e from both sides, we get C = 10 - e.
Finally, we substitute this value of C back into the equation to get the final solution: |x|y = eˣ + (10 - e).
So the solution to the initial-value problem is y = (eˣ + (10 - e))/|x|.