72.9.7 problem 7

Internal problem ID [14803]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.1. page 258
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 07:15:57 AM
CAS classification : system_of_ODEs

\begin{align*} p^{\prime }\left (t \right )&=3 p \left (t \right )-2 q \left (t \right )-7 r \left (t \right )\\ q^{\prime }\left (t \right )&=-2 p \left (t \right )+6 r \left (t \right )\\ r^{\prime }\left (t \right )&=\frac {73 q \left (t \right )}{100}+2 r \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.283 (sec). Leaf size: 912

dsolve([diff(p(t),t)=3*p(t)-2*q(t)-7*r(t),diff(q(t),t)=-2*p(t)+6*r(t),diff(r(t),t)=73/100*q(t)+2*r(t)],singsol=all)
 

\begin{align*} p &= \frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{{4}/{3}}-\left (31130+6 i \sqrt {895302429}\right )^{{4}/{3}}-96420 i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}-9395896 i \sqrt {3}+180 i \sqrt {895302429}-10228 \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}-540 \sqrt {298434143}+96420 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}-9395896\right ) c_{1} {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}-3214 i \sqrt {3}+\left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}-100 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}+3214\right ) t}{60 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}}}}{4800 \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}}-\frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{{4}/{3}}+\left (31130+6 i \sqrt {895302429}\right )^{{4}/{3}}-96420 i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}-9395896 i \sqrt {3}-180 i \sqrt {895302429}+10228 \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}-96420 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}-540 \sqrt {298434143}+9395896\right ) c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}-3214 i \sqrt {3}-\left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}+100 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}-3214\right ) t}{60 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}}}}{4800 \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}}-\frac {\left (-\left (31130+6 i \sqrt {895302429}\right )^{{4}/{3}}+180 i \sqrt {895302429}+5114 \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}+96420 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}-9395896\right ) c_{3} {\mathrm e}^{\frac {\left (\left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}+50 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}+3214\right ) t}{30 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}}}}{2400 \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}} \\ q &= c_{1} {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}-3214 i \sqrt {3}+\left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}-100 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}+3214\right ) t}{60 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}}}+c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}-3214 i \sqrt {3}-\left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}+100 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}-3214\right ) t}{60 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}}}+c_{3} {\mathrm e}^{\frac {\left (\left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}+50 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}+3214\right ) t}{30 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}}} \\ r &= \frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{{4}/{3}}-\left (31130+6 i \sqrt {895302429}\right )^{{4}/{3}}+32140 i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}-10641096 i \sqrt {3}-60 i \sqrt {895302429}-6228 \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}-32140 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}+180 \sqrt {298434143}-10641096\right ) c_{1} {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}-3214 i \sqrt {3}+\left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}-100 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}+3214\right ) t}{60 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}}}}{14400 \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}}-\frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{{4}/{3}}+\left (31130+6 i \sqrt {895302429}\right )^{{4}/{3}}+32140 i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}-10641096 i \sqrt {3}+60 i \sqrt {895302429}+6228 \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}+32140 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}+180 \sqrt {298434143}[phone]\right ) c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}-3214 i \sqrt {3}-\left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}+100 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}-3214\right ) t}{60 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}}}}{14400 \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}}+\frac {\left (\left (31130+6 i \sqrt {895302429}\right )^{{4}/{3}}+60 i \sqrt {895302429}-3114 \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}+32140 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}[phone]\right ) c_{3} {\mathrm e}^{\frac {\left (\left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}+50 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}+3214\right ) t}{30 \left (31130+6 i \sqrt {895302429}\right )^{{1}/{3}}}}}{7200 \left (31130+6 i \sqrt {895302429}\right )^{{2}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 602

DSolve[{D[p[t],t]==3*p[t]-2*q[t]-7*r[t],D[ q[t],t]==-2*p[t]+6*r[t],D[r[t],t]==73/100*q[t]+2*r[t]},{p[t],q[t],r[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} p(t)\to -100 c_2 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}[phone]\&,\frac {2 \text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}+111 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ]-100 c_3 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}[phone]\&,\frac {7 \text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}+1200 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}[phone]\&,\frac {\text {$\#$1}^2 e^{\frac {\text {$\#$1} t}{100}}-200 \text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}-43800 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ] \\ q(t)\to -200 c_1 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}[phone]\&,\frac {\text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}-200 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ]+200 c_3 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}[phone]\&,\frac {3 \text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}-200 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}[phone]\&,\frac {\text {$\#$1}^2 e^{\frac {\text {$\#$1} t}{100}}-500 \text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}+60000 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ] \\ r(t)\to -14600 c_1 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}[phone]\&,\frac {e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ]+73 c_2 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}[phone]\&,\frac {\text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}-300 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-500 \text {$\#$1}^2-23800 \text {$\#$1}[phone]\&,\frac {\text {$\#$1}^2 e^{\frac {\text {$\#$1} t}{100}}-300 \text {$\#$1} e^{\frac {\text {$\#$1} t}{100}}-40000 e^{\frac {\text {$\#$1} t}{100}}}{3 \text {$\#$1}^2-1000 \text {$\#$1}-23800}\&\right ] \\ \end{align*}