First slide

Types of solutions of differential equations

Question

$The solution of differential equation x d y (y^{2} e^{x y} + e^{x / y}) = y d x (e^{x / y} - y^{2} e^{x y}), is$

Moderate

A

$x y = \log (e^{x} + λ)$
B

$x^{2} / y = \log (e^{x / y} + λ)$
C

$x y = \log (e^{x / y} + λ)$
D

$x y^{2} = \log (e^{x / y} + λ)$

Solution

$\begin{array}{l} x d y (y^{2} e^{x y} + e^{x / y}) = y d x (e^{x / y} - y^{2} e^{x y}) \\ \Rightarrow e^{x / y} (x d y - y d x) = - e^{x y} (x d y + y d x) y^{2} \\ \Rightarrow e^{x / y} (\frac{y d x + x d y}{y^{2}}) = e^{x y} (x d y + y d x) \\ \Rightarrow \int e^{x / y} d (\frac{x}{y}) = \int e^{x y} d (x y) \\ \Rightarrow e^{x / y} + c = e^{x y} \\ \Rightarrow x y = \log (e^{x / y} + c) \end{array}$

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