Find the differential equation representing the family of curves y= ae^{2x}+5,where a is an arbitrary constant.

 

 

 

 
 
 
 
 

Answers (1)

given:  y= ae^{2x}+5
On differentiating w.r.t x, we get
\frac{dy}{dx}= ae^{2x}\left ( 2 \right )+0\left [ By\: chain\: rule \right ]
\Rightarrow \frac{dy}{dx}= 2ae^{2x}-\left ( i \right )
Again differentiating w.r.t  x,
\frac{d^{2}y}{dx^{2}}= 2\left ( ae^{2x}2 \right )
\Rightarrow \frac{d^{2}y}{dx^{2}}= \frac{2dy}{dx}-\left [ using\left ( i \right ) \right ]
\Rightarrow \frac{d^{2}y}{dx^{2}}-\frac{2dy}{dx}= 0 this is the required differential equation

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