Question

Problem: Solve the differential equation (dy)/(dx) = y(x - 1) with the given initial value y(0) = -3. Show all work.

Answer 1

To solve the given differential equation dy/dx = y(x - 1), we use separation of variables and integrate both sides. We use the initial condition y(0) = -3 to find the integration constant and solve for y, yielding y = -3e^((x^2)/2 - x) as the solution.

Step-by-step explanation:

To solve the differential equation dy/dx = y(x - 1) with the initial value y(0) = -3, we begin by separating variables. Follow these steps:

1. Move all terms involving y to one side of the equation, and all terms involving x to the other side, to get dy/y = (x - 1)dx.
2. Now integrate both sides of the equation to obtain ln|y| = (x^2/2) - x + C, where C is the integration constant.
3. Since we are given that y(0) = -3, we use this to find C. We substitute x = 0 and y = -3 into the integrated equation to solve for C.
4. We find that ln|y| = (x^2/2) - x + ln|3|.
5. To solve for y, we exponentiate both sides of the equation to get y = ±3e^((x^2)/2 - x).
6. Given that y(0) = -3, we choose the negative solution, so the final answer is y = -3e^((x^2)/2 - x).