Topics

16

Term Test 2 / Problem 1 (morning)

« on: November 19, 2019, 04:14:47 AM »

(a) Find the general solution of
$$
y''-y=\frac{12}{e^{t}+1}.
$$

(b) Find solution satisfying
$$y(0)=y'(0)=0.$$

17

Term Test 2 / Problem 1 (main sitting)

« on: November 19, 2019, 04:13:51 AM »

(a) Find the general solution of
$$
y''+4y=\frac{1}{\cos^2(t)},\qquad -\frac{\pi}{2}<t<\frac{\pi}{2}.
$$

(b) Find solution satisfying
$$y(0)=y'(0)=0.$$

18

Term Test 1 / Problem 4 (afternoon)

« on: October 23, 2019, 06:27:03 AM »

(a) Find the general solution for equation
\begin{equation*}
y'' +2y' +17 y =40 e^{x} +130\sin(4x) .
\end{equation*}
(b) Find solution, satisfying $y(0)=0$, $y'(0)=0$.

19

Term Test 1 / Problem 4 (noon)

« on: October 23, 2019, 06:26:03 AM »

(a) Find the general solution for equation
\begin{equation*}
y'' -8y' +25y =18e^{4x} +104\cos(3x) .
\end{equation*}
(b) Find solution, satisfying $y(0)=0$, $y'(0)=0$.

20

Term Test 1 / Problem 4 (morning)

« on: October 23, 2019, 06:25:04 AM »

(a) Find the general solution for equation
\begin{equation*}
y'' -6y' +25y =16e^{3x} +102\sin(x) .
\end{equation*}
(b) Find solution, satisfying $y(0)=0$, $y'(0)=0$

21

Term Test 1 / Problem 4 (main)

« on: October 23, 2019, 06:24:24 AM »

(a) Find the general solution for equation
\begin{equation*}
y'' -6y' +10y =2e^{3x} +39\cos(x) .
\end{equation*}
(b) Find solution, satisfying $y(0)=0$, $y'(0)=0$.

22

Term Test 1 / Problem 3 (afternoon)

« on: October 23, 2019, 06:13:03 AM »

(a) Find the general solution for equation
\begin{equation*}
y'' -5y'+6 y= 52\cos (2x).
\end{equation*}
(b) Find solution, satisfying $y(0)=0$, $y'(0)=0$.

23

Term Test 1 / Problem 3 (noon)

« on: October 23, 2019, 06:12:04 AM »

(a) Find the general solution for equation
\begin{equation*}
y'' -4y'+3 y= 96\sinh (x).
\end{equation*}
(b) Find solution, satisfying $y(0)=0$, $y'(0)=0$.

24

Term Test 1 / Problem 3 (morning)

« on: October 23, 2019, 06:11:02 AM »

(a) Find the general solution for equation
\begin{equation*}
y'' -6y'+8 y= 48\sinh (2x).
\end{equation*}
(b) Find solution, satisfying $y(0)=0$, $y'(0)=0$.

Hint: $\sinh(x)=\dfrac{e^x-e^{-x}}{2}$.

25

Term Test 1 / Problem 3 (main)

« on: October 23, 2019, 06:09:33 AM »

(a) Find the general solution for equation
Find the general solution for equation
\begin{equation*}
y'' -2y'-3y= 16\cosh (x).
\end{equation*}
 
(b) Find solution, satisfying $y(0)=0$, $y'(0)=0$.

26

Term Test 1 / Problem 2 (afternoon)

« on: October 23, 2019, 06:03:55 AM »

(a) Find Wronskian  $W(y_1,y_2)(x)$ of a fundamental set of solutions $y_1(x) , y_2(x)$ for ODE 
\begin{equation*}
(2x+1)x y''+(2x+2)y'-2y=0.
\end{equation*}
(b) Check that $y_1(x)=x+1$ is a solution and find another linearly independent solution.

(c) Write the general solution, and find solution such that ${y(-1)=1, y'(-1)=0}$.

27

Term Test 1 / Problem 2 (noon)

« on: October 23, 2019, 06:02:44 AM »

(a) Find Wronskian  $W(y_1,y_2)(x)$ of a fundamental set of solutions $y_1(x) , y_2(x)$ for ODE 
\begin{equation*}
\bigl(x\cos(x)-\sin(x)\bigr)y''+x\sin(x)y'-\sin(x)y=0.
\end{equation*}
(b) Check that $y_1(x)=x$ is a solution and find another linearly independent solution.

(c) Write the general solution, and find solution such that ${y(\pi)=\pi, y'(\pi)=0}$.

28

Term Test 1 / Problem 2 (morning)

« on: October 23, 2019, 06:00:35 AM »

(a) Find Wronskian  $W(y_1,y_2)(x)$ of a fundamental set of solutions $y_1(x) , y_2(x)$ for ODE 
\begin{equation*}
x y''-(2x+1)y'+(x+1)y=0.
\end{equation*}
(b) Check that $y_1(x)=e^x$ is a solution and find another linearly independent solution.

(c) Write the general solution, and find solution such that ${y(1)=0, y'(1)=e}$.

29

Term Test 1 / Problem 2 (main)

« on: October 23, 2019, 05:59:13 AM »

(a) Find Wronskian  $W(y_1,y_2)(x)$ of a fundamental set of solutions $y_1(x) , y_2(x)$ for ODE 
\begin{equation*}
x^2 y'' -2xy' + (x^2+2)y=0
\end{equation*}
(b) Check that $y_1(x)=x\cos(x)$ is a solution and find another linearly independent solution.

(c) Write the general solution, and find solution such that ${y(\frac{\pi}{2})=1, y'(\frac{\pi}{2})=0}$.

30

Term Test 1 / Problem 1 (afternoon)

« on: October 23, 2019, 05:57:42 AM »

(a) Find integrating factor and then a general solution of ODE
\begin{equation*}
-y^2\sin(xy) + \bigl(-xy \sin(xy)+2\cos(xy)+3y\bigr) y'=0
\end{equation*}

(b) Also, find a solution satisfying $y(\dfrac{\pi}{3})=1$.