7.17.15 problem 16 (a)

Internal problem ID [528]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.6 (Applications of Bessel functions). Problems at page 261
Problem number : 16 (a)
Date solved : Monday, January 27, 2025 at 02:54:33 AM
CAS classification : [[_Riccati, _special]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.145 (sec). Leaf size: 43

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

\[ y = -\frac {\left (-\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselY}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right )\right ) x}{-\operatorname {BesselJ}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+\operatorname {BesselY}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )} \]

✓ Solution by Mathematica

Time used: 0.304 (sec). Leaf size: 68

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

\[ y(x)\to -\frac {x^2 \operatorname {BesselJ}\left (-\frac {5}{4},\frac {x^2}{2}\right )-x^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {x^2}{2}\right )+\operatorname {BesselJ}\left (-\frac {1}{4},\frac {x^2}{2}\right )}{2 x \operatorname {BesselJ}\left (-\frac {1}{4},\frac {x^2}{2}\right )} \]