78.17.5 problem 4

Internal problem ID [18411]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 27. Series Solutions of First Order Equations. Problems at page 208
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 11:49:36 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=1+y^{2} \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 14

Order:=6; 
dsolve([diff(y(x),x)=1+y(x)^2,y(0) = 0],y(x),type='series',x=0);
 

\[ y = x +\frac {1}{3} x^{3}+\frac {2}{15} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 19

AsymptoticDSolveValue[{D[y[x],x]==1+y[x]^2,{y[0]==0}},y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {2 x^5}{15}+\frac {x^3}{3}+x \]