7.5.9 problem 12

Internal problem ID [2355]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 1.10. Page 80
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 05:34:25 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} y^{\prime }&={\mathrm e}^{\left (y-t \right )^{2}} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.203 (sec). Leaf size: 23

ode:=diff(y(t),t) = exp((y(t)-t)^2); 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = t +\operatorname {RootOf}\left (\int _{\textit {\_Z}}^{1}\frac {1}{-1+{\mathrm e}^{\textit {\_a}^{2}}}d \textit {\_a} +t \right ) \]

✓ Mathematica. Time used: 0.693 (sec). Leaf size: 256

ode=D[y[t],t]== Exp[(y[t]-t)^2]; 
ic={y[0]==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _0^t-\frac {e^{(y(t)-K[1])^2}}{-1+e^{(y(t)-K[1])^2}}dK[1]+\int _0^{y(t)}-\frac {e^{(t-K[2])^2} \int _0^t\left (\frac {2 e^{2 (K[2]-K[1])^2} (K[2]-K[1])}{\left (-1+e^{(K[2]-K[1])^2}\right )^2}-\frac {2 e^{(K[2]-K[1])^2} (K[2]-K[1])}{-1+e^{(K[2]-K[1])^2}}\right )dK[1]-\int _0^t\left (\frac {2 e^{2 (K[2]-K[1])^2} (K[2]-K[1])}{\left (-1+e^{(K[2]-K[1])^2}\right )^2}-\frac {2 e^{(K[2]-K[1])^2} (K[2]-K[1])}{-1+e^{(K[2]-K[1])^2}}\right )dK[1]-1}{-1+e^{(t-K[2])^2}}dK[2]=\int _0^1\frac {1}{-1+e^{K[2]^2}}dK[2],y(t)\right ] \]

✓ Sympy. Time used: 0.801 (sec). Leaf size: 27

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

\[ y{\left (t \right )} = - t + \int \limits ^{C_{2}} \frac {1}{\tanh {\left (\frac {r^{2}}{2} \right )}}\, dr - \int \limits ^{C_{2} + t} \frac {1}{\tanh {\left (\frac {r^{2}}{2} \right )}}\, dr + 1 \]