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### Messages - Victor Ivrii

Pages: 1 2 [3] 4 5 ... 166
31
##### 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 $y(0)=0$, $y'(0)=0$.

32
##### 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 $y(0)=0$, $y'(0)=0$.

33
##### 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 $y(0)=0$, $y'(0)=0$.

34
##### 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: $\sinh(x)=\dfrac{e^x-e^{-x}}{2}$.

35
##### 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 $y(0)=0$, $y'(0)=0$.

36
##### 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 ${y(-1)=1, y'(-1)=0}$.

37
##### 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 ${y(\pi)=\pi, y'(\pi)=0}$.

38
##### 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 ${y(1)=0, y'(1)=e}$.

39
##### 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 ${y(\frac{\pi}{2})=1, y'(\frac{\pi}{2})=0}$.

40
##### 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 $y(\dfrac{\pi}{3})=1$.

41
##### Term Test 1 / Problem 1 (noon)
« on: October 23, 2019, 05:54:36 AM »
(a) Find integrating factor and then a general solution of ODE
\begin{equation*}
\bigl(2y+y^2\sin(x)\bigr) + \bigl(\sin(2x)+2y\cos(x)\bigr) y'=0
\end{equation*}

(b) Also, find a solution satisfying $y(\dfrac{\pi}{4})=\sqrt{2}$.

42
##### Term Test 1 / Problem 1 (morning)
« on: October 23, 2019, 05:53:12 AM »
(a) Find integrating factor and then a general solution of ODE
\begin{equation*}
\bigl(-y\sin(x)+y^3\cos(x)\bigr) + \bigl(3\cos(x)+5y^2\sin(x)\bigr) y'=0
\end{equation*}

(b) Also, find a solution satisfying $y(\dfrac{\pi}{4})=\sqrt{2}$.

43
##### Term Test 1 / Problem 1 (main sitting)
« on: October 23, 2019, 05:51:05 AM »
(a) Find integrating factor and then a general solution of ODE
\begin{equation*}
\bigl(y +3 y^2e^{2x}\bigr) + \bigl(1+2ye^{2x}\bigr) y'=0
\end{equation*}

(b) Also, find a solution satisfying $y(0)=1$.

44
##### Term Test 1 / Please post solutions
« on: October 20, 2019, 06:25:41 PM »
You may post solutions to Test 1. All solutions posted before this announcement were removed (all of them seem to be  prepared in advance).
One user who was too smart for his/her own good (posted several times something like a single digit to stake out the lot) got a posting ban.

Please type sin, cos, log, ...  as \sin, \cos , \log , ... to produce $\sin (x)$,... (upright and with a proper horizontal spacing)

45
##### Chapter 3 / Re: Euler equations
« on: October 17, 2019, 12:37:51 AM »
Indeed, $y$ is the same same in the sense that $y[t]=y(x(t))$. However $\dfrac{dy}{dt}\ne \dfrac{dy}{dx} (x(t))$ but contains an extra factor.

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