If (a+bx)ey/x=x,then prove that x3d2y/dx2=(xdy/dx-y)2 - Maths - Differential Equations

Question

If (a+bx)ey/x=x,then prove that
x3d2
​y/dx​2=(xdy/dx-y)2

Rishabh Mittal · Accepted Answer

a+bxeyx=xTaking log on both sides, we getlog a+bxeyx=log xlog a+bx+log eyx=log xlog a+bx+yx=log x              since log e =1Differentiating both sides wrt x , we get 1a+bx b +dydx
x-yx2= 1xdydx x-yx2=1x-ba+bxdydx x-yx2=axa+bxdydx
x-y=axa+bxAgain differentiating wrt xd2ydx2
x+dydx-dydx=aa+bx-axba+bx2xd2ydx2=a2a+bx2multiplying both sides by x2x3d2ydx2=a2x2a+bx2x3d2ydx2=dydx
x-y2           since dydx
x-y=axa+bx Hence Proved

If (a+bx)ey/x=x,then prove that x3d2 ​y/dx​2=(xdy/dx-y)2.

If (a+bx)e^y/x=x,then prove that x³d² y/dx²=(xdy/dx-y)^2.