Please provide a solution of this questio as Ihave stuck to eliminate e^x

if y=e^x.tan^-1x then prove that (1+x²)d²y/dx² -2(1-x+x²)dy/dx +(1-x²)y=0

Question

Please provide a solution of this questio as Ihave stuck to
eliminate ex
if y=ex tan-1x then prove that
(1+x2)d2y/dx2
-2(1-x+x2)dy/dx +(1-x2)y=0

Priyanka Kedia · Accepted Answer

Recheck your question
y=extan-1xDifferentiating w r t x on both sides, we
have,dydx=ddxextan-1x =exddxtan-1x+tan-1xddxex =ex11+x2+tan-1xex
=ex11+x2+tan-1x⇒dydx=ex11+x2+tan-1xAgain differentiating w r
t x on both sides, we have,d2ydx2=ddxex11+x2+tan-1x
d2ydx2=ddxex11+x2+tan-1x =exddx11+x2+tan-1x+11+x2+tan-1xddxex
=ex-2x1+x22+11+x2+11+x2+tan-1xex =ex-2x+1+x21+x22+11+x2+tan-1xex
=ex-2x+1+x21+x22+11+x2+tan-1x =ex-2x+1+x2+1+x21+x22+tan-1x
=ex-2x+2+2x21+x22+tan-1x⇒d2ydx2=ex-2x+2+2x21+x22+tan-1x
Now,L H S =1+x2d2ydx2-21-x+x2dydx+1-x2y
=1+x2ex-2x+2+2x21+x22+tan-1x-21-x+x2ex11+x2+tan-1x+1-x2extan-1x
=1+x2ex-2x+2+2x21+x22+1+x2extan-1x-21-x+x2ex11+x2-21-x+x2extan-1x+1-x2extan-1x
=21-x+x2ex11+x2+1+x2extan-1x-21-x+x2ex11+x2-21-x+x2extan-1x+1-x2extan-1x
=extan-1x1+x2-21-x+x2+1-x2 =extan-1x2x-2x2 =R H S
Hence Proved

Please provide a solution of this questio as Ihave stuck to eliminate ex if y=ex.tan-1x then prove that (1+x2)d2y/dx2 -2(1-x+x2)dy/dx +(1-x2)y=0

Please provide a solution of this questio as Ihave stuck to eliminate e^x

if y=e^x.tan^-1x then prove that (1+x²)d²y/dx² -2(1-x+x²)dy/dx +(1-x²)y=0