Question

9. Superposition Principle: Let L[yl=y" + y + xy, yı(x)=sinx ^yz(x)=x.If L[y] x = xsinx and L(y2 x)=x²+1 then use the superposition principle to find a solution to the differential equation: L[y]=4x²+4-6 xsinx

Answer 1

Solution:-

Given that

Superposition principle:

Let

L = [y] =y"+y' + ry

Yi = sin

Y2 = 2

L[y].2) = L [sinc

$L[y_1](x)=(\sin x)'''+(\sin x)'+x(\sin x)$

$L[y_1](x)=-\cos x+\cos x+x\sin x$

$L[y_1](x)=x\sin x$

$L[y_2](x)=L[x]$

$L[y_2](x)=(x)'''+(x)'+x(x)$

$L[y_2](x)=1+x^2$

$L[y](x)=4x^2+4-6x\sin x$

$L[y](x)=4(1+x^2)-6(x\sin x)$

$L[y](x)=4L[y_2]-6L[y_1]$

$L[y]=L[4y_2-6y_1]$

$y=4y_2-6y_1$

So,

$y=4x-6\sin x$

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