Question

Cosy-xsiny(dy/dx)=sec^2 x​

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{Differential equation is}$

$\mathsf{cosy-x\,siny\,\dfrac{dy}{dx}=sec^2x}$

$\underline{\textbf{To find:}}$

$\textsf{Solution of the given differential eqation}$

$\underline{\textbf{Solution:}}$

$\mathsf{Considerr,}$

$\mathsf{cosy-x\,siny\,\dfrac{dy}{dx}=sec^2x}$

$\textsf{Take,}$

$\textsf{\boldmath t=x\,cosy }$

$\textsf{\boldmath \dfrac{dt}{dx}=x(-siny)\dfrac{dy}{dx}+cosy }$

$\textsf{\boldmath \dfrac{dt}{dx}=-x\,siny\,\dfrac{dy}{dx}+cosy }$

$\textsf{The given equation can be written as}$

$\mathsf{\dfrac{dt}{dx}=sec^2x}$

$\mathsf{dt=sec^2x\;dx}$

$\mathsf{Integrating,}$

$\mathsf{\displaystyle\int\,dt=\int\,sec^2x\;dx}$

$\mathsf{t=tanx+C}$

$\implies\boxed{\mathsf{x\,cosy=tanx+C}}$

$\textsf{which is the required solution}$

$\underline{\textbf{Find more:}}$

Solve x(x-y)dy+y^2dx=0​

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Answer 2

SOLUTION

TO EVALUATE

$\displaystyle \sf{ \cos y - x \sin y \frac{dy}{dx} = { \sec}^{2}x }$

EVALUATION

Here the given differential equation is

$\displaystyle \sf{ \cos y - x \sin y \frac{dy}{dx} = { \sec}^{2}x }$

We solve it as below

$\displaystyle \sf{ \cos y - x \sin y \frac{dy}{dx} = { \sec}^{2}x }$

$\displaystyle \sf{ \implies \: \cos y \: dx - x \sin y \: dy = { \sec}^{2}x \: dx}$

$\displaystyle \sf{ \implies \: d(x\cos y \: ) = { \sec}^{2}x \: dx}$

On integration we get

$\displaystyle \sf{ \int d(x\cos y \: ) = \int { \sec}^{2}x \: dx}$

$\displaystyle \sf{ \implies x\cos y \: = \tan x \: + c}$

Where C is integration constant

FINAL ANSWER

Hence the required solution is

$\boxed{ \: \: \displaystyle \sf{ x\cos y \: = \tan x \: + c} \: \: }$

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