78.18.4 problem 3

Internal problem ID [18416]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 28. Second Order Linear Equations. Ordinary Points. Problems at page 217
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 11:49:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }-y x&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 54

Order:=6; 
dsolve(diff(y(x),x$2)+diff(y(x),x)-x*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {1}{6} x^{3}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{30} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 63

AsymptoticDSolveValue[D[y[x],{x,2}]+D[y[x],x]-x*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {x^5}{120}-\frac {x^4}{24}+\frac {x^3}{6}+1\right )+c_2 \left (-\frac {x^5}{30}+\frac {x^4}{24}+\frac {x^3}{6}-\frac {x^2}{2}+x\right ) \]