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  • Help me solve this - Differential equations - JEE Main-6

Solution of diffrential equation \cos \frac{dy}{dx}+ (\sin y) x= x   is 

  • Option 1)

    e^{x^2/2}(\sin y-1)= c

  • Option 2)

    e^{x}(\sin y-1)= c

  • Option 3)

    e^{x^2}(\sin y-1)= c

  • Option 4)

    e^{x^2}(\sin y+1)= c

 

Answers (1)

As we have learned

Extended Form of linear Differential Equation -

Sometimes, a differential equation is not linear but it can be converted into a linear differential equation

-

 

 Let \sin y = f .\Rightarrow \cos y \frac{dy}{dx}= \frac{df}{dx}   

equation reduces to 

\frac{df}{dx=(x)t}=x

which is again comparable with \frac{dy}{dx} + py = Q

I.F = e^{xdx} = e^{x^2/2} 

multiplying both sides with integrating factor , we get 

e^{x^2/2}\cdot \frac{df}{dx} + t e^{x^2/2}\cdot x = x e^{x^2/2}

\Rightarrow d/dx (e^{x^2/2} \cdot t ) = xe^{x^2/2}

\Rightarrow \int d/dx (e^{x^2/2} \cdot t ) - \int xe^{x^2/2}dx=C

\Rightarrow e^{x^2/2}\cdot t- e^{x^2/2}= c

\Rightarrow e^{x^2/2}\cdot (f-1)= C\Rightarrow e^{x^2/2}(\sin y - 1 ) = C

 

 


Option 1)

e^{x^2/2}(\sin y-1)= c

Option 2)

e^{x}(\sin y-1)= c

Option 3)

e^{x^2}(\sin y-1)= c

Option 4)

e^{x^2}(\sin y+1)= c

Posted by

subam

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