50.18.10 problem 4(b)

Internal problem ID [8104]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.3. Second-Order Linear Equations: Ordinary Points. Page 169
Problem number : 4(b)
Date solved : Monday, January 27, 2025 at 03:43:49 PM
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*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 22

Order:=8; 
dsolve([diff(y(x),x$2)+diff(y(x),x)-x*y(x)=0,y(0) = 0, D(y)(0) = 1],y(x),type='series',x=0);
 

\[ y = x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{30} x^{5}+\frac {1}{90} x^{6}-\frac {1}{1680} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 47

AsymptoticDSolveValue[{D[y[x],{x,2}]+D[y[x],x]-x*y[x]==0,{y[0]==0,Derivative[1][y][0] ==1}},y[x],{x,0,"8"-1}]
 

\[ y(x)\to -\frac {x^7}{1680}+\frac {x^6}{90}-\frac {x^5}{30}+\frac {x^4}{24}+\frac {x^3}{6}-\frac {x^2}{2}+x \]