First slide

Types of solutions of differential equations

Question

$If x^{2} (x^{2} - 1) \frac{d y}{d x} + x (x^{2} + 1) y = x^{2} - 1 then y (x - \frac{1}{x}) - \ln x =$

Moderate

A

$c + \frac{1}{2 x^{2}}$
B

$c - \frac{1}{2 x^{2}}$
C

$c + \frac{1}{x^{2}}$
D

$c - \frac{1}{x^{2}}$

Solution

$\begin{array}{l} \frac{d y}{d x} + \frac{x^{2} + 1}{x (x^{2} - 1)} y = \frac{1}{x^{2}} \\ I. F = \exp \int \frac{(x^{2} + 1) d x}{x (x - 1) (x + 1)} = \frac{x^{2} - 1}{x} \\ Sol. Is \frac{y (x^{2} - 1)}{x} = \int \frac{x^{2} - 1}{x^{3}} d x = \ln x + \frac{1}{2 x^{2}} + c \\ \Rightarrow y (x - \frac{1}{x}) - \ln x = C + \frac{1}{2 x^{2}} \end{array}$

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