50.18.6 problem 1(f)

Internal problem ID [8100]
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 : 1(f)
Date solved : Monday, January 27, 2025 at 03:43:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 69

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

\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {1}{360} x^{6}-\frac {1}{840} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{60} x^{5}+\frac {1}{360} x^{6}+\frac {1}{840} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 84

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

\[ y(x)\to c_1 \left (-\frac {x^7}{840}-\frac {x^6}{360}+\frac {x^5}{60}-\frac {x^3}{6}+\frac {x^2}{2}+1\right )+c_2 \left (\frac {x^7}{840}+\frac {x^6}{360}-\frac {x^5}{60}+\frac {x^3}{6}-\frac {x^2}{2}+x\right ) \]