Question

Solve $\dfrac{dy}{dx} = \dfrac{x + e^x}{y}$.

Hint:

Always try to rewrite differential equations in recognizable derivative forms for easier integration.

Answer 1

Step 1: Rearranging.
We have: \[ \frac{dy}{dx} = \frac{x + e^x}{y} \] Multiply both sides by $y$: \[ y \frac{dy}{dx} = x + e^x \]

Step 2: Recognizing derivative form.
This suggests: \[ \frac{d}{dx}\left( \frac{y^2}{2} \right) = x + e^x \]

Step 3: Integration.
Integrating both sides w.r.t. $x$: \[ \frac{y^2}{2} = \int (x + e^x) \, dx \] \[ \frac{y^2}{2} = \frac{x^2}{2} + e^x + C \]

Step 4: Simplify.
\[ y^2 = x^2 + 2e^x + C' \]

Final Answer: \[ \boxed{y^2 = x^2 + 2e^x + C} \]