First slide

Types of solutions of differential equations

Question

A solution of the differential equation $y x^{y - 1} d x + x^{y} \ln x d y = 0$ , when $y (2) = 1$ is

Moderate

A

$y^{x} = 1$
B

$x^{y} + 1 = 0$
C

$\frac{1}{y} = \log_{2} x$
D

$x = \log_{2} y$

Solution

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

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