80.7.38 problem D 7

Internal problem ID [21357]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 7. System of first order equations. Excercise 7.6 at page 162
Problem number : D 7
Date solved : Sunday, October 12, 2025 at 05:51:28 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.091 (sec). Leaf size: 84

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

\begin{align*} \left \{x \left (t \right ) &= -\frac {{\mathrm e}^{-2 t} \sqrt {-2 \,{\mathrm e}^{2 t} \left (c_1 \,{\mathrm e}^{4 t}+{\mathrm e}^{2 t}-c_2 \right )}}{2}, x \left (t \right ) = \frac {{\mathrm e}^{-2 t} \sqrt {-2 \,{\mathrm e}^{2 t} \left (c_1 \,{\mathrm e}^{4 t}+{\mathrm e}^{2 t}-c_2 \right )}}{2}\right \} \\ \{y \left (t \right ) &= -x \left (t \right ) \left (\frac {d}{d t}x \left (t \right )\right )+2 t\} \\ \end{align*}

✓ Mathematica. Time used: 0.073 (sec). Leaf size: 817

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

\begin{align*} y(t)&\to \frac {1}{2} \left (4 t+\sqrt {(-1+8 c_1) \cosh ^2(2 (t-c_2))}\right )\\ x(t)&\to -\frac {\sqrt {\cosh ^2(2 (t-c_2)) \left (\tanh ^2(2 (t-c_2))+\sqrt {(-1+8 c_1) \tanh ^2(2 (t-c_2)) \text {sech}^2(2 (t-c_2))}-1\right )}}{\sqrt {2}}\\ y(t)&\to \frac {1}{2} \left (4 t+\sqrt {(-1+8 c_1) \cosh ^2(2 (t-c_2))}\right )\\ x(t)&\to \frac {\sqrt {\cosh ^2(2 (t-c_2)) \left (\tanh ^2(2 (t-c_2))+\sqrt {(-1+8 c_1) \tanh ^2(2 (t-c_2)) \text {sech}^2(2 (t-c_2))}-1\right )}}{\sqrt {2}}\\ y(t)&\to \frac {1}{2} \left (4 t+\sqrt {(-1+8 c_1) \cosh ^2(2 (t-c_2))}\right )\\ x(t)&\to -\frac {\sqrt {-\cosh ^2(2 (t-c_2)) \left (-\tanh ^2(2 (t-c_2))+\sqrt {(-1+8 c_1) \tanh ^2(2 (t-c_2)) \text {sech}^2(2 (t-c_2))}+1\right )}}{\sqrt {2}}\\ y(t)&\to \frac {1}{2} \left (4 t+\sqrt {(-1+8 c_1) \cosh ^2(2 (t-c_2))}\right )\\ x(t)&\to \frac {\sqrt {-\cosh ^2(2 (t-c_2)) \left (-\tanh ^2(2 (t-c_2))+\sqrt {(-1+8 c_1) \tanh ^2(2 (t-c_2)) \text {sech}^2(2 (t-c_2))}+1\right )}}{\sqrt {2}}\\ y(t)&\to 2 t-\frac {1}{2} \sqrt {(-1+8 c_1) \cosh ^2(2 (t+c_2))}\\ x(t)&\to -\frac {\sqrt {\cosh ^2(2 (t+c_2)) \left (\tanh ^2(2 (t+c_2))+\sqrt {(-1+8 c_1) \tanh ^2(2 (t+c_2)) \text {sech}^2(2 (t+c_2))}-1\right )}}{\sqrt {2}}\\ y(t)&\to 2 t-\frac {1}{2} \sqrt {(-1+8 c_1) \cosh ^2(2 (t+c_2))}\\ x(t)&\to \frac {\sqrt {\cosh ^2(2 (t+c_2)) \left (\tanh ^2(2 (t+c_2))+\sqrt {(-1+8 c_1) \tanh ^2(2 (t+c_2)) \text {sech}^2(2 (t+c_2))}-1\right )}}{\sqrt {2}}\\ y(t)&\to 2 t-\frac {1}{2} \sqrt {(-1+8 c_1) \cosh ^2(2 (t+c_2))}\\ x(t)&\to -\frac {\sqrt {-\cosh ^2(2 (t+c_2)) \left (-\tanh ^2(2 (t+c_2))+\sqrt {(-1+8 c_1) \tanh ^2(2 (t+c_2)) \text {sech}^2(2 (t+c_2))}+1\right )}}{\sqrt {2}}\\ y(t)&\to 2 t-\frac {1}{2} \sqrt {(-1+8 c_1) \cosh ^2(2 (t+c_2))}\\ x(t)&\to \frac {\sqrt {-\cosh ^2(2 (t+c_2)) \left (-\tanh ^2(2 (t+c_2))+\sqrt {(-1+8 c_1) \tanh ^2(2 (t+c_2)) \text {sech}^2(2 (t+c_2))}+1\right )}}{\sqrt {2}} \end{align*}

✗ Sympy

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

NotImplementedError :