First slide

Types of solutions of differential equations

Question

$The solution of e^{\frac{x (y^{2} - 1)}{y}} \{x y^{2} d y + y^{3} d x\} + {y d x - x d y} = 0 is$

Moderate

A

$e^{x y} + e^{\frac{x}{y}} + c = 0$
B

$e^{x y} - e^{\frac{x}{y}} + c = 0$
C

$e^{x y} + e^{\frac{y}{x}} + c = 0$
D

$e^{x y} - e^{\frac{y}{x}} + c = 0$

Solution

$\begin{array}{l} e^{\frac{x (y^{2} - 1)}{y}} y^{2} (x d y + y d x) + (y d x - x d y) = 0 \\ e^{x y} y^{2} (x d y + y d x) + e^{\frac{x}{y}} (y d x - x d y) = 0 \\ e^{x y} (x d y + y d x) + e^{\frac{x}{y}} d (\frac{x}{y}) = 0 \\ \Rightarrow e^{x y} + e^{\frac{x}{y}} + c = 0 \end{array}$

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