If the solution curve of the differential equation passes through the points and , then is equal to ______.

3 Explanation: The given differential equation is, It is a linear differential equation. &there4; I.F. = Put so &there4; I.F. = So, required solution is, \begin{aligned} & y{\sqrt{\log x}}=\int{\frac{\sqrt{\log x}}{x}}dx \\ & \log x=v\Longrightarrow{\frac{1}{x}}dx=dv \\ & \Rightarrow y\sqrt{\log x}=\int\sqrt{v}dv+\mathrm{C} \\ & \Rightarrow y\sqrt{\log x}=\frac{2v^{3/2}}{3}+\mathrm{C} \\ & \Rightarrow y\sqrt{\log x}=\frac{2}{3}(\log x)^{3/2}+\mathrm{C} \\ & \mathrm{Now,~this~curve~passes~through~}\left(e,\frac{4}{3}\right)\mathrm{~and~}(e^{4},\alpha) \\ & \therefore\frac{4}{3}\sqrt{\log e}=\frac{2}{3}(\log e)^{3/2}+\mathrm{C} \\ & \Rightarrow\mathrm{C}=\frac{4}{3}-\frac{2}{3}=\frac{2}{3} \\ & \mathrm{Also,}\alpha\sqrt{\log e^{4}}=\frac{2}{3}(\log e^{4})^{3/2}+\frac{2}{3} \\ & \Rightarrow2\alpha=\frac{2}{

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

If the solution curve of the differential equation $(y - 2 \log_e x)dx + (x \log_e x^2) dy = 0, x > 1$ passes through the points $(e, \frac{4}{3})$ and $(e^{4}, α)$ , then $α$ is equal to ______.

JEE 2023 — Mathematics PYQ

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