Question

Determine the solution of the differential equation:

$\frac{dy}{dx} = \frac{y}{1+x}$

Answer 1

y = C(1+x)

Answer 2

The differential equation is solved by separating variables, integrating both sides, and then solving for $y$.

1. Separate $y$ and $x$ terms: $\frac{dy}{y} = \frac{dx}{1+x}$.

2. Integrate both sides: $\int \frac{dy}{y} = \int \frac{dx}{1+x} \implies \ln|y| = \ln|1+x| + C'$.

3. Exponentiate to solve for $y$: $y = e^{\ln|1+x| + C'} = e^{C'} \cdot (1+x)$.

4. Let $e^{C'} = C$: $y = C(1+x)$.

The solution of the differential equation is $y = C(1+x)$.