14.4.12 problem 12

Internal problem ID [2530]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.10. Existence-uniqueness theorem. Excercises page 80
Problem number : 12
Date solved : Monday, January 27, 2025 at 05:59:13 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*}

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 23

dsolve([diff(y(t),t)=exp( (y(t)-t)^2 ),y(0) = 1],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.394 (sec). Leaf size: 256

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

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