69.1.135 problem 194
Internal
problem
ID
[14288]
Book
:
DIFFERENTIAL
and
INTEGRAL
CALCULUS.
VOL
I.
by
N.
PISKUNOV.
MIR
PUBLISHERS,
Moscow
1969.
Section
:
Chapter
8.
Differential
equations.
Exercises
page
595
Problem
number
:
194
Date
solved
:
Tuesday, January 28, 2025 at 06:25:46 AM
CAS
classification
:
[[_Riccati, _special]]
\begin{align*} y^{\prime }&=y^{2}+x \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=1 \end{align*}
✓ Solution by Maple
Time used: 0.166 (sec). Leaf size: 95
dsolve([diff(y(x),x)=y(x)^2+x,y(0) = 1],y(x), singsol=all)
\[
y = \frac {\left (-2 \,3^{{5}/{6}} \pi +3 \Gamma \left (\frac {2}{3}\right )^{2} 3^{{2}/{3}}\right ) \operatorname {AiryAi}\left (1, -x \right )+\operatorname {AiryBi}\left (1, -x \right ) \left (3 \,3^{{1}/{6}} \Gamma \left (\frac {2}{3}\right )^{2}+2 \pi 3^{{1}/{3}}\right )}{\left (-2 \,3^{{5}/{6}} \pi +3 \Gamma \left (\frac {2}{3}\right )^{2} 3^{{2}/{3}}\right ) \operatorname {AiryAi}\left (-x \right )+\operatorname {AiryBi}\left (-x \right ) \left (3 \,3^{{1}/{6}} \Gamma \left (\frac {2}{3}\right )^{2}+2 \pi 3^{{1}/{3}}\right )}
\]
✓ Solution by Mathematica
Time used: 1.074 (sec). Leaf size: 145
DSolve[{D[y[x],x]==y[x]^2+x,{y[0]==1}},y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {\sqrt [3]{3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \left (x^{3/2} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2 x^{3/2}}{3}\right )-x^{3/2} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2 x^{3/2}}{3}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2 x^{3/2}}{3}\right )\right )-2 x^{3/2} \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2 x^{3/2}}{3}\right )}{2 x \left (\operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (\frac {1}{3},\frac {2 x^{3/2}}{3}\right )-\sqrt [3]{3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2 x^{3/2}}{3}\right )\right )}
\]