What is the solution of the differential equation?
Direction: Consider the following for the two (02) items that follow : Let y\,dx+(x-y^{3})dy=0 be a differential equation.
- A. y^{4}+2x=c
- B. y^{4}+3x=c
- C. 2xy^{4}+x=c
- D. 4xy-y^{4}=c ✓
Correct Answer: D. 4xy-y^{4}=c
Explanation
We rewrite it as a linear differential equation in terms of x: \frac{dx}{dy} + \frac{1}{y}x = y^2. The integrating factor is \text{I.F.} = e^{\int \frac{1}{y} dy} = e^{\ln y} = y. Multiplying the entire equation by y gives y\frac{dx}{dy} + x = y^3, which is \frac{d}{dy}(xy) = y^3. Integrating both sides with respect to y, we get xy = \frac{y^4}{4} + C' \implies 4xy - y^4 = C.
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