if x=acostheta+bsintheta and y=asintheta-bcostheta then prove that y 2 d 2y/dx^2 xdy/dx + y = 0? How can I prove this math problem?

# if x=acostheta+bsintheta and y=asintheta-bcostheta then prove that y 2 d 2y/dx^2 xdy/dx + y = 0

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Given, x=acosθ+bsinθ & y=asinθ-bcosθ

then dx/dθ = -asinθ+bcosθ & dy/dθ = acosθ+bsinθ

dy/dx= (dy/dθ) * (dθ/dx)

= (acosθ+bsinθ)/(-(asinθ-bcosθ)) = x/-y

∴ dx/dy = -x/y

apply derviative

d^2y/dx^2 = 1/-y + -(-x)/y^2 * dy/dx

by solving we can get :

y^2 * d^2y/dx^2 -x*dy/dx +y =0

hence proved

You can also refer to the similar problems like:

If x+y+z=0 teh show that x^3+y^3 + z^3 =3xyz

https://techanswered.com/question/if-xyz0-then-show-that-x3y3z33xyz/