Solutions

1.
Solve ODE + Exact BITMAPSETProbSolvHint0.2309in0.2802in0inq1

y′′ -6y +5y = 0, Exact solution is: y$\displaystyle \left(\vphantom{
t}\right.$t$\displaystyle \left.\vphantom{
t}\right)$ = C1e5t + C2et

2.
Solve ODE + Exact (Specify x) BITMAPSETProbSolvHint0.2309in0.2802in0inq2

x2y′′ -3xy -6y = 0, Exact solution is: y$\displaystyle \left(\vphantom{ x}\right.$x$\displaystyle \left.\vphantom{ x}\right)$ = C1x2+$\scriptstyle \sqrt{{10}}$ + C2x2-$\scriptstyle \sqrt{{10}}$

3.
Solve ODE + Exact (Specify x) BITMAPSETProbSolvHint0.2309in0.2802in0inq3

2x2y = xy + 3y2, Exact solution is: $\displaystyle {\frac{{1}}{{y\left(
x\right) }}}$ = $\displaystyle {\frac{{3+C_{1}\sqrt{x}}}{{x}}}$

4.
Solve ODE + Laplace BITMAPSETProbSolvHint0.2309in0.2802in0inq4

y + y = 2
y(0) = 0
, Laplace solution is: y$\displaystyle \left(\vphantom{
t}\right.$t$\displaystyle \left.\vphantom{
t}\right)$ = 2 - 2e-t

5.
Solve ODE + Exact BITMAPSETProbSolvHint0.2309in0.2802in0inq5

$\displaystyle {\frac{{dy}}{{dt}}}$ - y - 3x = 0
-5y + $\displaystyle {\frac{{dx}}{{dt}}}$ - 3x = 0
y(0) = 1
x(0) = - 1
    Exact solution is: $\displaystyle \begin{array}{rl}
\QDATOP {} {}y\left( t\right) & =e^{-2t} \\
\QDATOP {} {}\;\,x\left( t\right) & =-e^{-2t}
\end{array}$

6.
Solve ODE + Exact

$\displaystyle {\frac{{dP}}{{dt}}}$ = kP(2000 - P), Exact solution is: $\displaystyle {\frac{{1}}{{P\left(
t\right) }}}$ = $\displaystyle {\frac{{1}}{{2000}}}$ + e-2000ktC1

Solve + Exact BITMAPSETProbSolvHint0.2309in0.2802in0inq6

$\displaystyle {\dfrac{{1}}{{500}}}$ = $\displaystyle {\dfrac{{1}}{{2000}}}$ + e-2000k$\scriptstyle \left(\vphantom{ 10}\right.$10$\scriptstyle \left.\vphantom{ 10}\right)$C1
$\displaystyle {\dfrac{{1}}{{300}}}$ = $\displaystyle {\dfrac{{1}}{{2000}}}$ + e-2000k$\scriptstyle \left(\vphantom{ 0}\right.$ 0$\scriptstyle \left.\vphantom{ 0}\right)$C1

Solution is: $\displaystyle \left\{\vphantom{ C_{1}=\frac{17}{6000},k=-\frac{1}{20000}\ln
\frac{9}{17}}\right.$C1 = $\displaystyle {\frac{{17}}{{6000}}}$, k = - $\displaystyle {\frac{{1}}{{20000}}}$ln$\displaystyle {\frac{{9}}{{17}}}$$\displaystyle \left.\vphantom{ C_{1}=\frac{17}{6000},k=-\frac{1}{20000}\ln
\frac{9}{17}}\right\}$

Define + New Definition

C1 = $\displaystyle {\frac{{17}}{{6000}}}$  
k = - $\displaystyle {\frac{{1}}{{20000}}}$ln$\displaystyle {\frac{{9}}{{17}}}$  
P$\displaystyle \left(\vphantom{
t}\right.$t$\displaystyle \left.\vphantom{
t}\right)$ = $\displaystyle \left(\vphantom{ \frac{1}{2000}+e^{-2000kt}C_{1}}\right.$$\displaystyle {\frac{{1}}{{2000}}}$ + e-2000ktC1$\displaystyle \left.\vphantom{ \frac{1}{2000}+e^{-2000kt}C_{1}}\right)^{{-1}}_{}$  

Evaluate

P$\displaystyle \left(\vphantom{
t}\right.$t$\displaystyle \left.\vphantom{
t}\right)$ = $\displaystyle {\frac{{6000}}{{3+17\exp \left( \frac{1}{10}\left( 2\ln 3-\ln
17\right) t\right) }}}$

Evaluate Numerically

P$\displaystyle \left(\vphantom{ 20}\right.$20$\displaystyle \left.\vphantom{ 20}\right)$ = 773

7.
Solve ODE + Exact, Simplify BITMAPSETProbSolvHint0.2309in0.2802in0inq7
y′′ = $\displaystyle \sqrt{{1+\left( y^{\prime }\right) ^{2}}}$  
y(0) = 1  
y(0) = 0  

Exact solution is:
y$\displaystyle \left(\vphantom{
t}\right.$t$\displaystyle \left.\vphantom{
t}\right)$ = cosh t cosh 0 - sinh t sinh  0  
  = cosh t  

8.
Solve ODE + Exact

$\displaystyle {\frac{{dT}}{{dt}}}$ = k$\displaystyle \left(\vphantom{ T-70}\right.$T - 70$\displaystyle \left.\vphantom{ T-70}\right)$, Exact solution is: T$\displaystyle \left(\vphantom{
t}\right.$t$\displaystyle \left.\vphantom{
t}\right)$ = 70 + ektC1

Solve + Exact BITMAPSETProbSolvHint0.2309in0.2802in0inq8

160 = 70 + ek$\scriptstyle \left(\vphantom{ 0}\right.$ 0$\scriptstyle \left.\vphantom{ 0}\right)$C1
120 = 70 + ek$\scriptstyle \left(\vphantom{ 10}\right.$10$\scriptstyle \left.\vphantom{ 10}\right)$C1
, Solution is: $\displaystyle \left\{\vphantom{ C_{1}=90,k=\frac{1}{10}\ln \frac{5}{9}}\right.$C1 = 90, k = $\displaystyle {\frac{{1}}{{10}}}$ln$\displaystyle {\frac{{5}}{{9}}}$$\displaystyle \left.\vphantom{ C_{1}=90,k=\frac{1}{10}\ln \frac{5}{9}}\right\}$

Define + New Definition

C1 = 90  
k = $\displaystyle {\frac{{1}}{{10}}}$ln$\displaystyle {\frac{{5}}{{9}}}$  

Simplify

T$\displaystyle \left(\vphantom{
t}\right.$t$\displaystyle \left.\vphantom{
t}\right)$ = 70 + ektC1

Evaluate

T$\displaystyle \left(\vphantom{
t}\right.$t$\displaystyle \left.\vphantom{
t}\right)$ = 70 + 90 exp$\displaystyle \left(\vphantom{ \frac{1}{10}\left( \ln 5-2\ln 3\right)
t}\right.$$\displaystyle {\frac{{1}}{{10}}}$$\displaystyle \left(\vphantom{ \ln 5-2\ln 3}\right.$ln 5 - 2 ln 3$\displaystyle \left.\vphantom{ \ln 5-2\ln 3}\right)$t$\displaystyle \left.\vphantom{ \frac{1}{10}\left( \ln 5-2\ln 3\right)
t}\right)$

Evaluate Numerically

T(20) = 97.8o

9.
Solve ODE + Numeric
${\dfrac{{dC}}{{dt}}}$ = - 0.2C + 0.0004CR
${\dfrac{{dR}}{{dt}}}$ = - 0.01CR + 0.001RV
${\dfrac{{dV}}{{dt}}}$ = - 0.001RV + 0.001V$\left(\vphantom{ 1000-V}\right.$1000 - V$\left.\vphantom{ 1000-V}\right)$
C(0) = 100
R(0) = 1000
V(0) = 1000

Plot 2D + ODE C, R, VBITMAPSETProbSolvHint0.2309in0.2802in0inq9

dtbpF3.0277in2.028in0ptplotode1.wmf

Observe what these functions predict for a longer term.dtbpF3.0277in2.028in0ptplotode2.wmf

10.
Define + New Definition

w = $\displaystyle {\frac{{1.62}}{{16}}}$

Define + New Definition BITMAPSETProbSolvHint0.2309in0.2802in0inq10

c = $\displaystyle {\frac{{1}}{{200}}}$w

Define + New Definition

g = 32

Define + New Definition

θ = $\displaystyle {\frac{{\pi }}{{16}}}$

Define + New Definition

k = c

Define + New Definition

z = 170

Solve ODE + Numeric
$\displaystyle {\frac{{dv}}{{dt}}}$ = - $\displaystyle {\frac{{gcv}}{{w}}}$  
$\displaystyle {\frac{{du}}{{dt}}}$ = - $\displaystyle {\frac{{g}}{{w}}}$$\displaystyle \left(\vphantom{ w-kv}\right.$w - kv$\displaystyle \left.\vphantom{ w-kv}\right)$  
$\displaystyle {\frac{{dx}}{{dt}}}$ = v  
$\displaystyle {\frac{{dy}}{{dt}}}$ = u  
v(0) = z cos2θ  
u(0) = z cosθsinθ  
x(0) = 0  
y(0) = 0  

Plot 2D + Parametric (x, y) dtbpF3.0277in2.028in0ptparaplot.wmf

11.
Solve PDE

2Dxu + 3Dxyu = xy2

Define + New Definition BITMAPSETProbSolvHint0.2309in0.2802in0inq11

F1(x)

Define + New Definition

F2(x)

Define + New Definition

u(x, y) = $\displaystyle {\frac{{1}}{{4}}}$x2y2 - $\displaystyle {\frac{{3}}{{4}}}$x2y + $\displaystyle {\frac{{9}}{{8}%
}}$x2 + F1(y) + e-2y/3F2(x)

Evaluate

2Dxu(x, y) + 3Dxyu(x, y)



Subsections