Questions: Find the solution of the differential equation that passes through the given point. dy/dx=x^3 y y=-2 e^x[?] /[]

Solution Steps

We are given the differential equation \(\frac{dy}{dx} = x^3 y\). We separate the variables: \(\frac{1}{y} dy = x^3 dx\)

Integrating both sides, we get \(\int \frac{1}{y} dy = \int x^3 dx\) \(\ln |y| = \frac{x^4}{4} + C\)

Exponentiating both sides, we get \( |y| = e^{\frac{x^4}{4} + C} = e^C e^{\frac{x^4}{4}} \) Let \( A = \pm e^C \). Then \( y = A e^{\frac{x^4}{4}} \)

The solution passes through the point \((0, -2)\). So, when \(x = 0\), \(y = -2\). \( -2 = A e^{\frac{0^4}{4}} = A e^0 = A \) So \(A = -2\).

The solution to the differential equation is \( y = -2 e^{\frac{x^4}{4}} \)

Final Answer The final answer is $\boxed{y=-2e^{x^4/4}}$