1yyy2Solution1yyy2 1yd2ydx2dydx2 let dydxv d2ydx2dvdx d2ydx2

(1+y)y\'\'=(y\')^2

(1+y)d²y_/dx²=(dy_/dx)²

let dy_/dx=v

d²y_/dx²=dv_/dx

d²y_/dx²=dv_/dy *dy_/dx

d²y_/dx²=dv_/dy *v

(1+y)y\'\'=(y\')²

(1+y)dv_/dy *v=(v)²

(1+y)dv_/dy =v

dv_/v =dy_/(1+y)

integrate on both sides

dv_/v =dy_/(1+y)

ln(v)=ln(1+y) +c

v=e^{ln(1+y) +c}

v=e^ln(1+y)e^c

v=C(1+y)

dy_/dx=C(1+y)

dy_/(1+y) =Cdx

integrate on both sides

dy_/(1+y) =Cdx

ln(1+y)=Cx +D

(1+y)=e^{Cx +D}

(1+y)=De^Cx

y =De^Cx -1 where C,D are constants

$(1+y)y\'\'=(y\')^2Solution(1+y)y\'\'=(y\')^2 (1+y)d2y/dx2=(dy/dx)2 let dy/dx=v d2y/dx2=dv/dx d2y/dx2=dv/dy *dy/dx d2y/dx2=dv/dy *v (1+y)y\'\'=(y\')2 (1+y)dv/dy$

1yyy2Solution1yyy2 1yd2ydx2dydx2 let dydxv d2ydx2dvdx d2ydx2

Solution

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