Toronto Math Forum
MAT244-2013S => MAT244 Math--Tests => Quiz 1 => Topic started by: Victor Ivrii on January 21, 2013, 11:01:25 AM
-
Please post solution:
Find the solution of the given initial value problem:
$$
ty' + 2y = \sin (t) \quad (t > 0),\qquad y(\pi/2) = 1.
$$
-
$$
y'+\frac{2}{t} y = \frac{\sin t}{t} \\
\mu y' + \frac{2 \mu}{t} y = \mu \frac{\sin t}{t} \\
\mu = t^2 \\
\frac{d}{dt} t^2 y = t \sin t \\
t^2 y = \int t \sin t dt \\
y = \frac{\sin t - t \cos t + C}{t^2}
$$
use initial value
$$
1 = \frac{4\left(1 - 0 + C\right)}{\pi^2}\\
C= \frac{\pi^2-4}{4}\\
y = \frac{\sin t - t \cos t + \frac{\pi^2-4}{4}}{t^2}\\
y = \frac{4\sin t - 4t \cos t + \pi^2-4}{4t^2}
$$