Question:
The general solution of the differential equation \(\frac{dy}{dx}+\frac{y}{x} = x^2+5\) is ....
The general solution of the differential equation \(\frac{dy}{dx}+\frac{y}{x} = x^2+5\) is ....
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Multiply by the integrating factor x to make the left side a perfect derivative.
Updated On: Oct 1, 2026
- \(\frac{x^4}{4}+\frac{5x^2}{2}-xy = c\)
- \(\frac{x^4}{4}-\frac{5x^2}{2}-xy = c\)
- \(\frac{x^4}{4}-\frac{5x^2}{2}+xy = c\)
- \(\frac{x^4}{4}+\frac{5x^2}{2}+xy = c\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept
The equation is linear: \(\dfrac{dy}{dx}+P\,y=Q\) with \(P=\dfrac1x\) and \(Q=x^2+5\).
Step 2: Integrating factor
\[ \text{I.F.}=e^{\int\frac1xdx}=e^{\ln x}=x \]
Step 3: Multiply through
\[ x\frac{dy}{dx}+y=x^3+5x\ \Rightarrow\ \frac{d}{dx}(xy)=x^3+5x \]
Step 4: Integrate
\[ xy=\frac{x^4}{4}+\frac{5x^2}{2}+c' \]
Step 5: Rearrange
\[ \frac{x^4}{4}+\frac{5x^2}{2}-xy=c \]
This is option (A). Options with a minus sign on \(5x^2/2\) have the wrong sign from the integral of \(5x\).
Final Answer:
The integrating factor is x and the solution is x^4/4 + 5x^2/2 - xy = c, option (A). \[ \boxed{\frac{x^4}{4}+\frac{5x^2}{2}-xy=c} \]
The equation is linear: \(\dfrac{dy}{dx}+P\,y=Q\) with \(P=\dfrac1x\) and \(Q=x^2+5\).
Step 2: Integrating factor
\[ \text{I.F.}=e^{\int\frac1xdx}=e^{\ln x}=x \]
Step 3: Multiply through
\[ x\frac{dy}{dx}+y=x^3+5x\ \Rightarrow\ \frac{d}{dx}(xy)=x^3+5x \]
Step 4: Integrate
\[ xy=\frac{x^4}{4}+\frac{5x^2}{2}+c' \]
Step 5: Rearrange
\[ \frac{x^4}{4}+\frac{5x^2}{2}-xy=c \]
This is option (A). Options with a minus sign on \(5x^2/2\) have the wrong sign from the integral of \(5x\).
Final Answer:
The integrating factor is x and the solution is x^4/4 + 5x^2/2 - xy = c, option (A). \[ \boxed{\frac{x^4}{4}+\frac{5x^2}{2}-xy=c} \]
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