First slide

Methods of solving first order first degree differential equations

Question

Solution to the differential equation $\frac{x + \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + ....}{1 + \frac{x^{2}}{2!} + \frac{x^{4}}{4!} ....} = \frac{d x - d y}{d x + d y}$

Moderate

A

$2 y e^{2 x} = c e^{2 x} + 1$
B

$2 y e^{2 x} = c e^{2 x} - 1$
C

$2 y e^{2 x} = c e^{2 x} + 2$
D

none of these

Solution

Given equation can be written as
$\frac{\sinh x}{\cosh x} = \frac{1 - \frac{d y}{d x}}{1 + \frac{d y}{d x}}$
Apply componendo and dividendo
$\begin{array}{l} \frac{\sinh x + \cosh x}{\sinh x - \cosh x} = \frac{2}{- 2 \frac{d y}{d x}} \\ \Rightarrow \frac{e^{x}}{- e^{- x}} = \frac{d x}{- d y} \\ \Rightarrow d y = e^{- 2 x} d x \end{array}$
Integrate we get
$\begin{array}{l} y = \frac{e^{- 2 x}}{- 2} + \frac{c}{2} \Rightarrow - 2 y e^{2 x} = 1 - c e^{2 x} \\ \Rightarrow 2 y e^{2 x} = c e^{2 x} - 1 \end{array}$

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