631
Quiz 3 / Day Section Problem 1
« on: February 27, 2013, 07:46:14 PM »
Find the general solution of
\begin{equation*}
y'''-y''+y'-y=e^{-t}\sin(t).
\end{equation*}
\begin{equation*}
y'''-y''+y'-y=e^{-t}\sin(t).
\end{equation*}
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632
Quiz 3 / Night Sections Problem 2
« on: February 27, 2013, 07:45:30 PM »
4.3 p 239, # 4
Find a particular solution and then the general solution of the following ODE
\begin{equation*}
y'''-y'= 2 \sin t .
\end{equation*}
Find a particular solution and then the general solution of the following ODE
\begin{equation*}
y'''-y'= 2 \sin t .
\end{equation*}
633
Quiz 3 / Night Sections Problem 1
« on: February 27, 2013, 07:43:47 PM »
4.4 p 244, # 1
Find a particular solution by method of variation of parameters and then the general solution of the following ODE :
\begin{equation*}
y'''+y'=\tan t,\qquad -\frac{\pi}{2}<t<\frac{\pi}{2}.
\end{equation*}
Find a particular solution by method of variation of parameters and then the general solution of the following ODE :
\begin{equation*}
y'''+y'=\tan t,\qquad -\frac{\pi}{2}<t<\frac{\pi}{2}.
\end{equation*}
634
Term Test 1 / Official solutions
« on: February 21, 2013, 04:47:31 PM »
There are "official" solutions
635
Reading Week Challenge / Reading Week Bonus problem 4
« on: February 16, 2013, 10:40:07 AM »
Using Reading Week Bonus problem 3 find inequalities for two consecutive zeros $x_n$ and $x_{n+1}$ of Airy function satisfying equation
\begin{equation}
y''+xy=0,
\end{equation}
Then derive asymptotic formula for $x_n$ as $n\to +\infty$.
\begin{equation}
y''+xy=0,
\end{equation}
Then derive asymptotic formula for $x_n$ as $n\to +\infty$.
636
Reading Week Challenge / Reading Week Bonus problem 3
« on: February 16, 2013, 10:37:44 AM »
Using Reading Week Bonus problem 2 consider $Q$ as a constant which is the maximum (minimum) of $q(x)$ on interval $[x_n,x_{n+1}]$ where $x_n$ and $x_{n+1}$ are two consecutive zeros of $y(x)$ and estimate $(x_{n+1}-x_n)$ from above (from below--respectively).
637
Reading Week Challenge / Reading Week Bonus problem 2
« on: February 16, 2013, 10:35:08 AM »
Using Reading Week Bonus problem 1 prove (in its framework) that if $y(x_0)=y(x_1)=0$ and $y(x)>0$ on $(x_0,x_1)$, $x_0<x_1$ then $z(x)$ has a $0$ somewhere on $(x_0,x_1)$ unless $z(x_0)=y(x_1)=0$ and $Q(x)=q(x)$ on $(x_0,x_1)$.
638
Reading Week Challenge / Reading Week Bonus problem 1
« on: February 16, 2013, 10:27:54 AM »
For "theoretical dudes" I suggest the sequence of the problems devoted comparison theorems and studying consecutive zeroes of solutions to 2nd order linear homogeneous equations.
Consider two equations:
\begin{gather}
y'' + q(x)y=0,\\
z''+Q(x)z''=0
\end{gather}
with
\begin{equation}
Q(x)\ge q(x).
\end{equation}
Consider Wronskian $W(x):=W[y,z](x)$ and assuming that $y(x)>0$, $z(x)>0$ on interval $[a,b]$ derive differential inequality
\begin{equation}
W' ? 0.
\end{equation}
Consider two equations:
\begin{gather}
y'' + q(x)y=0,\\
z''+Q(x)z''=0
\end{gather}
with
\begin{equation}
Q(x)\ge q(x).
\end{equation}
Consider Wronskian $W(x):=W[y,z](x)$ and assuming that $y(x)>0$, $z(x)>0$ on interval $[a,b]$ derive differential inequality
\begin{equation}
W' ? 0.
\end{equation}
640
Term Test 1 / TT1--Problem 4
« on: February 13, 2013, 10:41:28 PM »
Find solution
\begin{equation*}
y^{(4)}+8y''+16y=0
\end{equation*}
satisfying initial conditions
\begin{equation*}
y(0)=1,\; y'(0)=y''(0)=y'''(0)=0.
\end{equation*}
\begin{equation*}
y^{(4)}+8y''+16y=0
\end{equation*}
satisfying initial conditions
\begin{equation*}
y(0)=1,\; y'(0)=y''(0)=y'''(0)=0.
\end{equation*}
641
Term Test 1 / TT1--Problem 3
« on: February 13, 2013, 10:39:58 PM »
Find the general solution for equation
\begin{equation*}
y'' + 4y'+5y =t e^{-2t}+ e^{-2t}\cos(t).
\end{equation*}
\begin{equation*}
y'' + 4y'+5y =t e^{-2t}+ e^{-2t}\cos(t).
\end{equation*}
642
Term Test 1 / TT1--Problem 2
« on: February 13, 2013, 10:38:31 PM »
(a) Consider equation
\begin{equation*}
(\cos(t)+t\sin(t))y''-t\cos(t)y'+y\cos(t)=0.
\end{equation*}
Find wronskian $W=W[y_1,y_2](t)$ of two solutions such that $W(0)=1$.
(b) Check that one of the solutions is $y_1(t)=t$. Find another solution $y_2$ such that $W[y_1,y_2](\pi/2)=\pi/2$
and $y_2(\pi/2)=0$.
\begin{equation*}
(\cos(t)+t\sin(t))y''-t\cos(t)y'+y\cos(t)=0.
\end{equation*}
Find wronskian $W=W[y_1,y_2](t)$ of two solutions such that $W(0)=1$.
(b) Check that one of the solutions is $y_1(t)=t$. Find another solution $y_2$ such that $W[y_1,y_2](\pi/2)=\pi/2$
and $y_2(\pi/2)=0$.
643
Term Test 1 / TT1--Problem 1
« on: February 13, 2013, 10:36:58 PM »
Find integrating factor and solve
\begin{equation*}
x\,dx +y (1+x^2+y^2)\,dy=0.
\end{equation*}
\begin{equation*}
x\,dx +y (1+x^2+y^2)\,dy=0.
\end{equation*}
644
Term Test 1 / Posting solutions
« on: February 13, 2013, 04:55:52 PM »
About 22:30 today I will post problems--and then you post solutions. Please note that not only solutions but also discussion of solutions are welcome.
645
Ch 4 / Bonus problem for week 5b
« on: February 07, 2013, 11:54:56 PM »
Write down an $m$-th order homogeneous linear equation with constant coefficients (with the smallest possible $m$) such that it has solutions
\begin{equation*}
y_1= e^t, \qquad y_2= te^{-t}.
\end{equation*}
\begin{equation*}
y_1= e^t, \qquad y_2= te^{-t}.
\end{equation*}