JEE 2024 — Mathematics PYQ

\begin{aligned}
 & \text{Given:D.E.is} \\
 & (2x+3y-2)dx+(4x+6y-7)dy=0 \\
 & \Rightarrow\frac{dy}{dx}=\frac{-2x+3y-2}{4x+6y-7} & ...(\mathbf{i}) \\
 & \operatorname{Let}2x+3y=\mu \\
 & \Rightarrow2+3\frac{dy}{dx}=\frac{du}{dx} \\
 & \text{So, (i) becomes} \\
 & \Rightarrow\frac{1}{3}\left(\frac{du}{dx}-2\right)=-\frac{u-2}{24-7} \\
 & \mathrm{=}\Rightarrow{\frac{du}{dx}}={\frac{-3u+6}{24-7}}+2={\frac{-3u+6+4u-14}{2u-7}} \\
 & \Rightarrow\frac{du}{dx}=\frac{4-8}{24-7} \\
 & \Rightarrow\int\frac{24-7}{4-8}du=\int dx \\
 & \Rightarrow\int\left[1+\frac{4+1}{4-8}\right]du=\int dx
\end{aligned}