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  • Consider the differential equation <img alt="\frac{dy}{dx}=\frac{y^{3}}{2(xy^{2}-x^{2})}" src="https://learn.careers360.com/latex-image/?%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%7By%5E%7B3%7D%7D

Consider the differential equation

\frac{dy}{dx}=\frac{y^{3}}{2(xy^{2}-x^{2})} :

Statement 1 : The substitution z=y^{2} transforms the above equation into a first order homogenous differential equation.

Statement 2 : The solution of this differential equation is y^{2}e^{\frac{-y^{2}}{x}}=C

Option: 1

Both statements are false 


Option: 2

Statement 1 is true and Statement 2 is false 


Option: 3

Statement 1 is false and Statement 2 is true 


Option: 4

Both Statements are true 


Answers (1)

best_answer

\frac{dy}{dx}=\frac{y^{3}}{2(xy^{2}-x^{2})}

\\z=y^2\\\frac{dz}{dx}=2y\frac{dy}{dx}\Rightarrow \frac{dy}{dx}=\frac{1}{2y}\frac{dz}{dx}\\

\\\frac{1}{2y}\frac{dz}{dx}=\frac{yz}{2(xy^2-x^2)}\\\frac{dz}{dx}=\frac{z^2}{(xz^2-x^2)}

this is homogeneous form

Now, consider y^{2}e^{\frac{-y^{2}}{x}}=C

Differentiate both sides wrt x

\\2y\frac{dy}{dx}\cdot e^{-\frac{y^2}{x}}+y^2\cdot e^{-\frac{y^2}{x}}(-1)\left ( \frac{x\cdot2y\frac{dy}{dx}-y^2}{x^2} \right )=0\\ e^{-\frac{y^2}{x}}\left ( 2y\frac{dy}{dx}+\frac{y^2}{x^2}\left ( y^2-2xy\frac{dy}{dx} \right ) \right )=0\\2y\frac{dy}{dx}+\frac{y^4}{x^2}-\frac{2y^3}{x}\frac{dy}{dx}=0\\2x^2y\frac{dy}{dx}+y^4-{2xy^3}\frac{dy}{dx}=0

\\2x^2\frac{dy}{dx}+y^3-{2xy^2}\frac{dy}{dx}=0

 

 

 

 

 

Posted by

Ritika Harsh

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