If x = sint and y = sinpt prove that :

(1 - xsquare)d2y/dx2 - xdy/dx + psquarey = 0

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If x = sint and y = sinpt prove that :
(1 - xsquare)d2y/dx2 - xdy/dx + psquarey = 0

Vipin Verma · Accepted Answer

As x = sint, y = sinptThese have parameters tSo dydx=dydtdxdtHence
dydt=pcospt, dxdt=costSo dydx=dydtdxdt=pcosptcost (i)And for
d2ydx2, differentiate both sides of (i), wrt x and multiplying by
dtdx on the RHSAs dydx is in the form of uv, so
d2ydx2=vu'-uv'v2Hence
d2ydx2=cost(-p2sinpt)-pcospt(-sint)cos2t×dtdx=-p2costsinpt +
pcosptsintcos2t×1cost=-p2costsinpt +
pcosptsint1-sin2t×1costHence (1-x2)d2ydx2-xdydx+p2y
=(1-sin2t)d2ydx2-sintdydx+p2sinpt=(1-sin2t)×-p2costsinpt +
pcosptsint1-sin2t×1cost-sint×pcosptcost+p2sinpt=-p2sinpt+sint×pcosptcost-sint×pcosptcost+p2sinpt=0
(hence proved)

If x = sint and y = sinpt prove that : (1 - xsquare)d2y/dx2 - xdy/dx + psquarey = 0

If x = sint and y = sinpt prove that :

(1 - xsquare)d2y/dx2 - xdy/dx + psquarey = 0