51.1.2 problem 1. direct method
Internal
problem
ID
[8213]
Book
:
A
course
in
Ordinary
Differential
Equations.
by
Stephen
A.
Wirkus,
Randall
J.
Swift.
CRC
Press
NY.
2015.
2nd
Edition
Section
:
Chapter
8.
Series
Methods.
section
8.2.
The
Power
Series
Method.
Problems
Page
603
Problem
number
:
1.
direct
method
Date
solved
:
Monday, January 27, 2025 at 03:46:05 PM
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: 0.126 (sec). Leaf size: 89
dsolve([diff(y(x),x)=y(x)^2-x,y(0) = 1],y(x), singsol=all)
\[
y = \frac {-2 \pi \operatorname {AiryAi}\left (1, x\right ) 3^{{5}/{6}}-3 \operatorname {AiryAi}\left (1, x\right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{{2}/{3}}-3 \operatorname {AiryBi}\left (1, x\right ) 3^{{1}/{6}} \Gamma \left (\frac {2}{3}\right )^{2}+2 \pi \operatorname {AiryBi}\left (1, x\right ) 3^{{1}/{3}}}{2 \pi \operatorname {AiryAi}\left (x \right ) 3^{{5}/{6}}+3 \operatorname {AiryAi}\left (x \right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{{2}/{3}}+3 \operatorname {AiryBi}\left (x \right ) 3^{{1}/{6}} \Gamma \left (\frac {2}{3}\right )^{2}-2 \pi \operatorname {AiryBi}\left (x \right ) 3^{{1}/{3}}}
\]
✓ Solution by Mathematica
Time used: 0.660 (sec). Leaf size: 164
DSolve[{D[y[x],x]==y[x]^2-x,{y[0]==1}},y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {\sqrt [3]{-3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \left (i x^{3/2} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} i x^{3/2}\right )-i x^{3/2} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} i x^{3/2}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i x^{3/2}\right )\right )-2 i x^{3/2} \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2}{3} i x^{3/2}\right )}{2 x \left (\operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {BesselJ}\left (\frac {1}{3},\frac {2}{3} i x^{3/2}\right )-\sqrt [3]{-3} \operatorname {Gamma}\left (\frac {2}{3}\right ) \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} i x^{3/2}\right )\right )}
\]