Question

Solve the differential equation $\displaystyle \frac{dy}{dx} + \frac{y}{x} = x^2$.

Hint:

Remember: Linear DE → Use integrating factor $e^{\int P(x)dx}$.

Answer 1

Concept: This is a linear differential equation of the form: \[ \frac{dy}{dx} + P(x)y = Q(x) \] where $P(x) = \frac{1}{x}$ and $Q(x) = x^2$. Step 1: {Find the integrating factor (I.F.).}
\[ \text{I.F.} = e^{\int P(x)\,dx} = e^{\int \frac{1}{x}dx} = e^{\ln x} = x \]
Step 2: {Multiply both sides by I.F.}
\[ x \frac{dy}{dx} + y = x^3 \]
Step 3: {Recognize LHS as derivative.}
\[ \frac{d}{dx}(xy) = x^3 \]
Step 4: {Integrate both sides.}
\[ xy = \int x^3 dx = \frac{x^4}{4} + C \]
Step 5: {Solve for $y$.}
\[ y = \frac{x^3}{4} + \frac{C}{x} \]
Step 6: {Conclusion.}
\[ \boxed{y = \frac{x^3}{4} + \frac{C}{x}} \]