Find the solution of the differential equation that satisfies the given initial condition.dydx = xy, y(0) = −1

Sure, let's solve this differential equation step by step.

1. The given differential equation is dy/dx = xy. This is a first order linear differential equation.

2. To solve this, we can use the method of separation of variables. We separate the variables x and y on two sides of the equation. So, we get dy/y = x dx.

3. Now, we integrate both sides of the equation. The integral of dy/y is ln|y| and the integral of x dx is (1/2)x^2. So, we get ln|y| = (1/2)x^2 + C, where C is the constant of integration.

4. To get y, we take the exponential of both sides. This gives us |y| = e^(1/2)x^2+C = e^(1/2)x^2 * e^C. We can write e^C as another constant, say A. So, we get |y| = A * e^(1/2)x^2.

5. Since y can be positive or negative, we get y = ± A * e^(1/2)x^2.

6. Now, we use the initial condition y(0) = -1. Substituting x = 0 in the equation, we get -1 = ± A * e^0. This gives A = -1.

7. So, the solution of the differential equation is y = - e^(1/2)x^2.