Question:
The solution of the differential equation \(x\frac{dy}{dx}+y = x^3y^6\), is...
The solution of the differential equation \(x\frac{dy}{dx}+y = x^3y^6\), is...
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This is a Bernoulli equation; substitute v = 1/y^5.
Updated On: Oct 1, 2026
- \(y^5 = \frac{5}{2}x^{-3}+cx^5\)
- \(y^5 = \frac{5}{2}x^3+cx^5\)
- \(\frac{1}{y^5} = \frac{5}{2}x^{-3}+cx^5\)
- \(\frac{1}{y^5} = \frac{5}{2}x^3+cx^5\)
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The Correct Option is D
Solution and Explanation
Step 1: Rewrite in standard form:
Divide by \(x\): \(\dfrac{dy}{dx}+\dfrac yx=x^2y^6\). This is a Bernoulli equation with \(n=6\).
Step 2: Substitute:
Divide by \(y^6\): \(y^{-6}y'+\dfrac1x y^{-5}=x^2\). Let \(v=y^{-5}\), so \(v'=-5y^{-6}y'\). Then \(-\dfrac{v'}{5}+\dfrac vx=x^2\), that is \(v'-\dfrac5xv=-5x^2\).
Step 3: Solve the linear equation:
Integrating factor: \(e^{\int-5/x\,dx}=x^{-5}\). Then \(\dfrac{d}{dx}(vx^{-5})=-5x^{-3}\), so \(vx^{-5}=\dfrac52x^{-2}+c\).
Step 4: Return to y:
\(v=\dfrac52x^3+cx^5\), so \(\dfrac1{y^5}=\dfrac52x^3+cx^5\). Option D.
Step 5: Why the other options are wrong.
Options A and B give \(y^5\) instead of \(1/y^5\) because they forgot that \(v=y^{-5}\). Option C has \(x^{-3}\) from a sign slip in the integrating factor.
Final Answer:
The solution is 1/y^5 = (5/2)x^3 + c x^5. \[ \boxed{\text{(D) }\dfrac{1}{y^5}=\dfrac52x^3+cx^5} \]
Divide by \(x\): \(\dfrac{dy}{dx}+\dfrac yx=x^2y^6\). This is a Bernoulli equation with \(n=6\).
Step 2: Substitute:
Divide by \(y^6\): \(y^{-6}y'+\dfrac1x y^{-5}=x^2\). Let \(v=y^{-5}\), so \(v'=-5y^{-6}y'\). Then \(-\dfrac{v'}{5}+\dfrac vx=x^2\), that is \(v'-\dfrac5xv=-5x^2\).
Step 3: Solve the linear equation:
Integrating factor: \(e^{\int-5/x\,dx}=x^{-5}\). Then \(\dfrac{d}{dx}(vx^{-5})=-5x^{-3}\), so \(vx^{-5}=\dfrac52x^{-2}+c\).
Step 4: Return to y:
\(v=\dfrac52x^3+cx^5\), so \(\dfrac1{y^5}=\dfrac52x^3+cx^5\). Option D.
Step 5: Why the other options are wrong.
Options A and B give \(y^5\) instead of \(1/y^5\) because they forgot that \(v=y^{-5}\). Option C has \(x^{-3}\) from a sign slip in the integrating factor.
Final Answer:
The solution is 1/y^5 = (5/2)x^3 + c x^5. \[ \boxed{\text{(D) }\dfrac{1}{y^5}=\dfrac52x^3+cx^5} \]
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