problem (a)

79.10.1 problem (a)

Internal problem ID [21108]
Book : Ordinary Diﬀerential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter IV. Linear Diﬀerential Equations. Excercises V at page 173
Problem number : (a)
Date solved : Sunday, October 12, 2025 at 05:51:26 AM
CAS classiﬁcation : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=\left (3 t -1\right ) x \left (t \right )-\left (1-t \right ) y \left (t \right )+t \,{\mathrm e}^{t^{2}}\\ \frac {d}{d t}y \left (t \right )&=-\left (t +2\right ) x \left (t \right )+\left (t -2\right ) y \left (t \right )-{\mathrm e}^{t^{2}} \end{align*}

✓ Maple. Time used: 0.124 (sec). Leaf size: 115

ode:=[diff(x(t),t) = (3*t-1)*x(t)-(1-t)*y(t)+t*exp(t^2), diff(y(t),t) = -(t+2)*x(t)+(t-2)*y(t)-exp(t^2)]; 
dsolve(ode);

\begin{align*} x \left (t \right ) &= \left (c_2 +\frac {1}{9} t^{3}+\frac {1}{9} t^{2}+\frac {4}{9} t +\frac {4}{9}\right ) {\mathrm e}^{t^{2}}+\left (3 \,{\mathrm e}^{t^{2}-3 t} t -2 \,{\mathrm e}^{t^{2}-3 t}\right ) c_1 \\ y \left (t \right ) &= -\frac {{\mathrm e}^{t^{2}} t^{3}}{9}-3 \,{\mathrm e}^{t^{2}-3 t} c_1 t -\frac {t^{2} {\mathrm e}^{t^{2}}}{9}-7 \,{\mathrm e}^{t^{2}-3 t} c_1 -{\mathrm e}^{t^{2}} c_2 -\frac {t \,{\mathrm e}^{t^{2}}}{9}-\frac {8 \,{\mathrm e}^{t^{2}}}{9} \\ \end{align*}

✓ Mathematica. Time used: 0.475 (sec). Leaf size: 196

ode={D[x[t],t]==(3*t-1)*x[t]-(1-t)*y[t]+t*Exp[t^2], D[y[t],t]==-(t+2)*x[t]+(t-2)*y[t]-Exp[t^2]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]

\begin{align*} x(t)&\to \frac {c_1 e^{t^2-\frac {1}{2}} \sqrt {t-1}}{\sqrt {2-2 t}}+\frac {\sqrt {2} c_2 e^{t^2-3 t+\frac {5}{2}} \sqrt {t-1} (3 t-2)}{9 \sqrt {1-t}}+\frac {1}{81} e^{t^2} \left (9 t^3+9 t^2+36 t-8\right )\\ y(t)&\to \frac {1}{162} e^{t^2} \left (-2 \left (9 t^3+9 t^2+9 t+28\right )-\frac {81 \sqrt {\frac {2}{e}} c_1 (t-1)}{\sqrt {-(t-1)^2}}-\frac {18 \sqrt {2} c_2 e^{\frac {5}{2}-3 t} (t-1) (3 t+7)}{\sqrt {-(t-1)^2}}\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-t*exp(t**2) + (1 - t)*y(t) - (3*t - 1)*x(t) + Derivative(x(t), t),0),Eq((2 - t)*y(t) + (t + 2)*x(t) + exp(t**2) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)

NotImplementedError :

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