Second Order Differential Equations Find the general solutio

Second Order Differential Equations

Find the general solution of the equation (dy/dt)^ 2 = A^2 k ^2y^ 2 . Assume A > 0, k > 0. (Suggestion: begin by taking the derivative of this equation with respect to t.) This problem shows that you can get oscillations from a first order DE if it is nonlinear

Solution

GIven ode is

(y\')^2=A^2-k^2y^2

Differentiating w.r.t. t gives

2y\'y\'\'=-2k^2yy\'

y\'y\'\'+k^2yy\'=0

y\'(y\'\'+k^2y)=0\\\\

y\'=0, y\'\'+k^2y=0

Setting y\'=0 in original give ode gives

A^2-k^2y^2=0

y=A/k

SEcond eqution is

y\'\'+k^2y=0

General solution to this ode is

y=C sin(kt)+D cos(kt)

HEnce general solution to given equation is

y= C sin(kt)+D cos(kt)+A/k