MHTCET 16 Apr 2026 Shift 1 Paper

Concept:The given differential equation is linear in $x$ if we treat $x$ as a function of $y$, so we solve it using an integrating factor.Explanation:Given:$$\left(x+2y^{3}\right)\frac{dy}{dx}-y=0$$Rearrange to get:$$\left(x+2y^{3}\right)\frac{dy}{dx}=y$$Dividing by $y$, we can write:$$\frac{dx}{dy}=\frac{x+2y^{3}}{y}$$This simplifies to a linear equation in $x$:$$\frac{dx}{dy}-\frac{x}{y}=2y^{2}$$The integrating factor (I.F.) is:$$\text{I.F.}=e^{\int-\frac{1}{y}\,dy}=\frac{1}{y}$$Multiplying both sides by $\frac{1}{y}$, the left side becomes the derivative of $\frac{x}{y}$:$$\frac{d}{dy}\left(\frac{x}{y}\right)=2y$$Integrating both sides with respect to $y$:$$\frac{x}{y}=y^{2}+c$$Hence, the required solution is:$$x=\left(c+y^{2}\right)y$$Answer:Therefore, the correct option is A: $x=\left(c+y^{2}\right)y$, where $c$ is the constant of integration.