77.1.34 problem 51 (page 75)

Internal problem ID [17924]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 51 (page 75)
Date solved : Tuesday, January 28, 2025 at 11:12:49 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 45

dsolve(diff(y(x),x)=y(x)^2-x^2,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.123 (sec). Leaf size: 196

DSolve[D[y[x],x]==y[x]^2-x^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {i x^2 \left (2 \operatorname {BesselJ}\left (-\frac {3}{4},\frac {i x^2}{2}\right )+c_1 \left (\operatorname {BesselJ}\left (-\frac {5}{4},\frac {i x^2}{2}\right )-\operatorname {BesselJ}\left (\frac {3}{4},\frac {i x^2}{2}\right )\right )\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )}{2 x \left (\operatorname {BesselJ}\left (\frac {1}{4},\frac {i x^2}{2}\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )\right )} \\ y(x)\to -\frac {i x^2 \operatorname {BesselJ}\left (-\frac {5}{4},\frac {i x^2}{2}\right )-i x^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {i x^2}{2}\right )+\operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )}{2 x \operatorname {BesselJ}\left (-\frac {1}{4},\frac {i x^2}{2}\right )} \\ \end{align*}