Question

The solution of the differential equation $x \frac{dy}{dx} + 2y = x^2$ is:

• A. $y x^2 = \frac{x^4}{4} + C$
• B. $y x = \frac{x^3}{3} + C$
• C. $y x^2 = \frac{x^3}{3} + C$
• D. $y x = \frac{x^4}{4} + C$

Hint:

Differentiating the LHS of the solution $y x^2$ yields $x^2 \frac{dy}{dx} + 2xy$, which is precisely $x$ times the LHS of the original differential equation. This is a quick way to verify the integrating factor.

Answer 1

The Correct Option is A

Step 1: Rewrite in standard form.
The equation $x\frac{dy}{dx} + 2y = x^2$ should look like $\frac{dy}{dx} + Py = Q$. Divide every term by $x$.
\[ \frac{dy}{dx} + \frac{2}{x}y = x \] So $P = \frac{2}{x}$ and $Q = x$.

Step 2: Find the integrating factor.
The integrating factor is $e^{\int P\,dx}$. Here $\int\frac{2}{x}\,dx = 2\ln|x| = \ln x^2$.
\[ \text{I.F.} = e^{\ln x^2} = x^2 \]

Step 3: Write the solution form.
The solution is $y$ times the integrating factor equals the integral of $Q$ times the integrating factor.
\[ y\,x^2 = \int x\cdot x^2\,dx + C \]

Step 4: Simplify the integrand.
Multiply $x$ by $x^2$ to get $x^3$.
\[ y\,x^2 = \int x^3\,dx + C \]

Step 5: Do the integral.
The integral of $x^3$ is $\frac{x^4}{4}$.
\[ y\,x^2 = \frac{x^4}{4} + C \]

Step 6: State the answer.
This is the general solution.
\[ \boxed{y\,x^2 = \tfrac{x^4}{4} + C} \]