14.4.3 problem 3

Internal problem ID [2521]
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 : 3
Date solved : Monday, January 27, 2025 at 05:58:27 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&={\mathrm e}^{t}+y^{2} \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.180 (sec). Leaf size: 63

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

\[ y = \frac {\left (-\operatorname {BesselJ}\left (1, 2 \,{\mathrm e}^{\frac {t}{2}}\right ) \operatorname {BesselY}\left (1, 2\right )+\operatorname {BesselY}\left (1, 2 \,{\mathrm e}^{\frac {t}{2}}\right ) \operatorname {BesselJ}\left (1, 2\right )\right ) {\mathrm e}^{\frac {t}{2}}}{\operatorname {BesselJ}\left (1, 2\right ) \operatorname {BesselY}\left (0, 2 \,{\mathrm e}^{\frac {t}{2}}\right )-\operatorname {BesselJ}\left (0, 2 \,{\mathrm e}^{\frac {t}{2}}\right ) \operatorname {BesselY}\left (1, 2\right )} \]

✓ Solution by Mathematica

Time used: 0.282 (sec). Leaf size: 79

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

\[ y(t)\to \frac {\sqrt {e^t} \left (\operatorname {BesselJ}(1,2) \operatorname {BesselY}\left (1,2 \sqrt {e^t}\right )-\operatorname {BesselY}(1,2) \operatorname {BesselJ}\left (1,2 \sqrt {e^t}\right )\right )}{\operatorname {BesselJ}(1,2) \operatorname {BesselY}\left (0,2 \sqrt {e^t}\right )-\operatorname {BesselY}(1,2) \operatorname {BesselJ}\left (0,2 \sqrt {e^t}\right )} \]