Final answer:
To solve the initial value problem e¹y' = e¹ + 3x with y(2) = 5, variables are separated and both sides of the equation are integrated. The resulting antiderivatives are used to solve for y, then the initial condition is applied to find the constant and get the specific solution.
Step-by-step explanation:
The initial value problem posed is e¹y' = e¹ + 3x, with the initial condition y(2) = 5. To solve this, we need to separate the variables and integrate both sides. First, we rewrite the equation as y' = 1 + 3xe-y. Then we separate variables by moving all y terms to one side and x terms to the other, yielding dy / (1 + 3xe-y) = dx. Now, we can integrate both sides. Let's denote the integration of the left side as F(y) and the integration of the right side as G(x). After finding the antiderivatives, we can then solve for y. Finally, we apply the initial value y(2) = 5 to determine the constant of integration to find the particular solution to the differential equation.