solve yy y2 using the substitution uy and UdudyySolutionyy

solve yy\'\' = (y\')^2 using the substitution u=y\' and U(du/dy)=y\'\'

Solution

yy\'\' = (y\')^2

Substituting u=y\' and u(du/dy)=y\'\',

y((u(du/dy)) = u²

y ( du/dy ) = u

du/u = dy/y

Integrating both the sides of the above equation, we get

ln|u| = ln|y| + c₁ which gives u =c₂y

Resubstituting u = dy/dx, we get

dy/dx = c₂y

dy/y = c₂ dx

Integrating both the sides of the above equation, we get

ln|y| = c₂ x + c₃ or y = c₄ e^c₂x