Question

If $y=y(x)$ is the solution of the differential equation \[ x\frac{dy}{dx}+2y=x^2 \] satisfying $y(1)=1$, then $y\!\left(\dfrac{1}{2}\right)$ is equal to:

• A. $\dfrac{1}{4}$
• B. $\dfrac{7}{64}$
• C. $\dfrac{49}{16}$
• D. $\dfrac{13}{16}$

Hint:

First-order linear differential equations are best solved using the integrating factor method.

Answer 1

The Correct Option is D

Step 1: Rewrite the given differential equation in standard linear form: \[ \frac{dy}{dx}+\frac{2}{x}y=x \]
Step 2: Identify the integrating factor (I.F.): \[ \text{I.F.}=e^{\int \frac{2}{x}\,dx}=e^{2\ln x}=x^2 \]
Step 3: Multiply the entire equation by the integrating factor: \[ x^2\frac{dy}{dx}+2xy=x^3 \]
Step 4: Observe that the left-hand side is the derivative of $x^2y$: \[ \frac{d}{dx}(x^2y)=x^3 \]
Step 5: Integrate both sides: \[ x^2y=\int x^3\,dx=\frac{x^4}{4}+C \]
Step 6: Solve for $y$: \[ y=\frac{x^2}{4}+\frac{C}{x^2} \]
Step 7: Use the condition $y(1)=1$: \[ 1=\frac{1}{4}+C \Rightarrow C=\frac{3}{4} \]
Step 8: Substitute $C=\frac{3}{4}$: \[ y=\frac{x^2}{4}+\frac{3}{4x^2} \]
Step 9: Evaluate $y\!\left(\dfrac{1}{2}\right)$: \[ y\!\left(\frac{1}{2}\right) =\frac{(1/2)^2}{4}+\frac{3}{4(1/2)^2} =\frac{1}{16}+3 =\frac{13}{16} \]