First slide

Methods of solving first order first degree differential equations

Question

The equation of the curve satisfying the differential equation $y^{2} (x^{2} + 1) = 2 x \frac{d y}{d x}$ passing through the point (1,1) is

Moderate

A

$y = \frac{4}{5 + x + \ln x}$
B

$y = \frac{4}{5 - x + 2 \ln x}$
C

$y = \frac{4}{5 - x^{2} - 2 \ln x}$
D

$y = \frac{4}{5 - x^{2} + 2 \ln x}$

Solution

Given differential equation after rearranging the terms is
$\begin{array}{l} \int \frac{x^{2} + 1}{2 x} d x = \int \frac{d y}{y^{2}} \\ \frac{1}{2} (\int x d x + \int \frac{1}{x} d x) = \int \frac{d y}{y^{2}} \\ \Rightarrow \frac{1}{2} (\frac{x^{2}}{2} + \ln x) = \frac{- 1}{y} + C \\ ∵ It passes through (1, 1) \end{array}$
$\begin{array}{l} \Rightarrow \frac{1}{4} = - 1 + C \Rightarrow C = \frac{5}{4} \\ \Rightarrow \frac{1}{y} = \frac{5}{4} - (\frac{x^{2} + 2 \ln x}{4}) \\ \Rightarrow y = \frac{4}{5 - x^{2} - 2 \ln x} \end{array}$

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