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  • Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b. y = e^x(a cosx + b sinx)

5.Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

        y=e^x(a\cos x + b\sin x)

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Given equation is
y=e^x(a\cos x + b\sin x)                  -(i)
Now, differentiate w.r.t x
\frac{dy}{dx}= \frac{d(e^{x}(a\cos x+b\sin x))}{dx}= e^{x}(a\cos x+b\sin x)+e^x(-a\sin x+b\cos x )           -(ii)
Now, again differentiate w.r.t x
y^{''}= \frac{d^2y}{dx^2}= \frac{d}{dx}\frac{dy}{dx} =e^{x}(a\cos x+b\sin x)+e^x(-a\sin x+b\cos x )+e^x(-a\sin x+b\cos x )+e^x(-a\cos x-b\sin x)                  
                                         =2e^x(-a\sin x+b\cos x )          -(iii)
Now, multiply equation (i) with 2  and   multiply equation (ii) with 2 and add and subtract from  equation (iii) respectively
we will get

 y^{''}-2y^{'}+2y = 0
Therefore, the required equation is  y^{''}-2y^{'}+2y = 0

Posted by

Gautam harsolia

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