Question

The solution of \[ x\frac{dy}{dx}+y=x^4 \] with \[ y(1)=\frac{6}{5} \] is

• A. \(y=\dfrac{x^4}{5}+1\)
• B. \(y=\dfrac{x^4}{5}+\dfrac{1}{x}\)
• C. \(y=\dfrac{4x^4}{5}+\dfrac{4}{5x}\)
• D. \(y=\dfrac{x}{5}+1\)

Hint:

Whenever you see \(x\dfrac{dy}{dx}+y\), recognize it as \(\dfrac{d}{dx}(xy)\).

Answer 1

The Correct Option is B

Concept:
The given differential equation is a first order linear differential equation. \[ x\frac{dy}{dx}+y=x^4 \] Observe that the left side resembles the derivative of \(xy\), because \[ \frac{d}{dx}(xy)=x\frac{dy}{dx}+y \] 

Step 1: Rewrite the equation. \[ x\frac{dy}{dx}+y=x^4 \] Using the product derivative, \[ \frac{d}{dx}(xy)=x^4 \] 

Step 2: Integrate both sides. \[ \int \frac{d}{dx}(xy)\,dx=\int x^4\,dx \] \[ xy=\frac{x^5}{5}+C \] 

Step 3: Divide by \(x\). \[ y=\frac{x^4}{5}+\frac{C}{x} \] 

Step 4: Use the condition \(y(1)=\dfrac{6{5}\).} Substitute \(x=1\) and \(y=\dfrac{6}{5}\): \[ \frac{6}{5}=\frac{1^4}{5}+\frac{C}{1} \] \[ \frac{6}{5}=\frac{1}{5}+C \] \[ C=\frac{6}{5}-\frac{1}{5} \] \[ C=1 \] 

Step 5: Write the final solution. \[ y=\frac{x^4}{5}+\frac{1}{x} \] 

Step 6: Final answer. \[ \boxed{y=\frac{x^4}{5}+\frac{1}{x}} \]