14.4.4 problem 4

Internal problem ID [2522]
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 : 4
Date solved : Monday, January 27, 2025 at 05:58:30 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+\cos \left (t \right )^{2} \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.392 (sec). Leaf size: 16

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

\[ y = -\frac {\operatorname {MathieuCPrime}\left (\frac {1}{2}, -\frac {1}{4}, t\right )}{\operatorname {MathieuC}\left (\frac {1}{2}, -\frac {1}{4}, t\right )} \]

✓ Solution by Mathematica

Time used: 0.214 (sec). Leaf size: 25

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

\[ y(t)\to -\frac {\text {MathieuCPrime}\left [\frac {1}{2},-\frac {1}{4},t\right ]}{\text {MathieuC}\left [\frac {1}{2},-\frac {1}{4},t\right ]} \]