Q4 TUT 0601

[Victor Ivrii]

Find the general solution of the given differential equation.
$$
y'' + y = \tan (t),\qquad 0< t < \frac{\pi}{2}.
$$

[Jiabei Bi]

here is my solution

[Jiacheng Ge]

For the homogeneous equation,
r² + 1 = 0
r = i or −i

So, the complementary solution is y = C1cost + C2sint

W[y1,y2] =y1y2' − y1'y2 = cos²t + sin²t = 1

u1(x)=−∫(y2(x)g(x)/W)dx
        = −∫(sint tant) dt
        = sint − ln(tant+sect)

u2(x)=∫(y1(x)g(x)/W)dx
        = ∫(cost tant)dt
        = ∫sint dt
        = −cos t

So, a particualr solution is
y = u1y1+u2y2
   =[sint - ln(tant+sect)]cost − costsint
   =−ln(tant+sect)cost

So,the general solution is
y = C1cost + C2sint −ln(tant+sect)cost
                                       

[Zhiya Lou]

First, solve the homogeneous solution:
$r^2+1=0$
$r_1=i, r_2= -i$
$y_1= cos(t);  y_2 = sin(t)$
So, $W=y_1y_2' - y_2y_1'= cos^2(t) + sin^2(t) = 1$

Second, solve for particular solution:
$Y=u_1y_1 + u_2y_2$
$u_1(t) = -\int sin(t)tan(t) dt\ = -\int sec(t)(1-cos^2(t))dt = -\int sec(t) - cos(t) dt = -ln(tan(t) + sec(t)) +sin(t) $

$u_2(t) = \int y_1(t) tan(t) dt = \int cos(t)tan(t) dt = \int sin(t) dt = -cos(t)$

Therefore, $Y= [-ln(tan(t) + sec(t)) +sin(t)](cos(t)) + (-cos(t)) (sin(t))$


General Solution:
$y(t)= c_1cos(t) + c_2sin(t) - ln(tan(t)+sec(t))(cos(t))$

[Victor Ivrii]

Jiacheng, unreadable

Zhiya
Need to type \sin x, \ln x ...