Question

Solve the differential equation. $$ y^{\prime}=x(1+y) $$

Short answer

The solutions for y are \(y = e^{\frac{1}{2}x^2 + C} - 1\) or \(y = 1 - e^{\frac{1}{2}x^2 + C}\)

Step-by-step solution

Rewrite in Separable Form

Rearrange the original equation to put all terms involving y on one side and all terms involving x on the other: \(\frac{dy}{dx} = x(1+y) \Rightarrow \frac{1}{1+y} dy = x dx\).

Integrate Both Sides

Apply the integral to both sides of the equation: \(\int\frac{1}{1+y} dy = \int x dx\).

Solve the Integrals

Solve the integrals on both sides: \(\ln|1+y| = \frac{1}{2}x^2 + C\).

Solve for y

To isolate y, first exponentiate both sides to remove the natural log: \(|1+y| = e^{\frac{1}{2}x^2 + C}\). Because the absolute value can be positive or negative, this gives us two possible values for 1+y. Solving for y gives us: \(y = e^{\frac{1}{2}x^2 + C} - 1\) or \(y = 1 - e^{\frac{1}{2}x^2 + C}\).