Answer to Question #297347 in Differential Equations for John

Question

{F} 2. Find the particular solution of the differential equation 1 − cos(2t) dx/dt = x sin(2t) given that x(π/2) = 2.

Answer 1

Let us find the particular solution of the differential equation $(1 − \cos(2t))\frac{ dx}{dt} = x \sin(2t)$ given that $x(\frac{π}2) = 2.$

This equation is equivalent to $\frac{ dx}{x} = \frac{\sin(2t)dt}{1 − \cos(2t)}$, and hence $\int\frac{ dx}{x} = \int\frac{\sin(2t)dt}{1 − \cos(2t)}$ .

It follows that

$\ln|x|= \frac{1}2\int\frac{d(1 − \cos(2t))}{1 − \cos(2t)}=\frac{1}2\ln|1 − \cos(2t)|+\ln|C|=\ln|C\sqrt{1 − \cos(2t)}|.$

We conclude that $x(t)=C\sqrt{1 − \cos(2t)}$ is the general solution.

If $x(\frac{π}2) = 2$ then $2=C\sqrt{1-\cos\pi}=C\sqrt{2}.$ It follows that $C=\sqrt{2},$ and thus the particular solution is

$x(t)=\sqrt{2}\sqrt{1 − \cos(2t)}=\sqrt{2 − 2\cos(2t)}.$