Question

X=asin(y+b) defferential equation

Answer 1

Question :-

Solve differential equation.

x = a sin (y +b)

Answer :-

$\rightarrow \red{ \boxed{ \sf{\frac{ {d}^{2} y}{d {x}^{2} } = x { \bigg( \frac{dy}{dx} \bigg)}^{2} }}} \:$

Explanation :-

We have ,

→ x = a sin (y +b)

This equation has two constants ( a and b) so that's why we will have to do differentiation two times ,

$\rightarrow \: \sf{ x = \red{ a } \purple{\sin( \green{y + b})} }\\ \\ \sf{ \purple{differentiate }\: with \: \red{ respect \:} to \: x} \\ \\ \rightarrow \sf{1 = \blue{ a} \red{ \cos(y + b)} \frac{d}{dx} \green{(y + b)}} \\ \\ \sf{ \rightarrow \: 1 = a \cos(y + b) \: \frac{dy}{dx} }....(1) \\ \\ \sf{ \green{again \: } \red{differentiate }\: it}$

differentiate it by product rule of differentiation ,

$\rightarrow \sf{0 = a \cos(y + b) \frac{ {d}^{2}y }{d {x}^{2} } + \frac{dy}{dx} \frac{d}{dx} a \cos(y + b)} \\ \\ \rightarrow \sf{ \green{- a \cos(y + b) \frac{ {d}^{2} y}{d {x}^{2} } } = \red{ - a \sin(y + b) \frac{dy}{dx} \times \frac{dy}{dx} }} \: \: \\ \sf{from \: (1)}\\ \\ \rightarrow \sf{ \purple{ - \frac{dx}{dy} \frac{ {d}^{2} y}{d {x}^{2} }} = \green{ - x { \bigg(\frac{dy}{dx} \bigg)}^{2} }} \\ \\ \rightarrow \red{ \boxed{ \sf{\frac{ {d}^{2} y}{d {x}^{2} } = x { \bigg( \frac{dy}{dx} \bigg)}^{2} }}}$

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