Internal problem ID [2794]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 2
Problem number: 39.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-x^{2}+y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 44
dsolve(diff(y(x),x) = x^2-y(x)^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {x \left (\BesselI \left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) c_{1}-\BesselK \left (\frac {3}{4}, \frac {x^{2}}{2}\right )\right )}{c_{1} \BesselI \left (\frac {1}{4}, \frac {x^{2}}{2}\right )+\BesselK \left (\frac {1}{4}, \frac {x^{2}}{2}\right )} \]
✓ Solution by Mathematica
Time used: 0.136 (sec). Leaf size: 103
DSolve[y'[x]==x^2 - y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {i x \left (J_{-\frac {3}{4}}\left (\frac {i x^2}{2}\right )-c_1 J_{\frac {3}{4}}\left (\frac {i x^2}{2}\right )\right )}{J_{\frac {1}{4}}\left (\frac {i x^2}{2}\right )+c_1 J_{-\frac {1}{4}}\left (\frac {i x^2}{2}\right )} \\ y(x)\to \frac {x I_{\frac {3}{4}}\left (\frac {x^2}{2}\right )}{I_{-\frac {1}{4}}\left (\frac {x^2}{2}\right )} \\ \end{align*}