Internal problem ID [5679]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\left (1+x \right ) y^{\prime }-y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 64
Order:=8; dsolve(diff(y(x),x$2)+(1+x)*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {1}{360} x^{6}-\frac {1}{840} x^{7}\right ) y \relax (0)+\left (x -\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{60} x^{5}+\frac {1}{360} x^{6}+\frac {1}{840} x^{7}\right ) D\relax (y )\relax (0)+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 84
AsymptoticDSolveValue[y''[x]+(1+x)*y'[x]-y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {x^7}{840}-\frac {x^6}{360}+\frac {x^5}{60}-\frac {x^3}{6}+\frac {x^2}{2}+1\right )+c_2 \left (\frac {x^7}{840}+\frac {x^6}{360}-\frac {x^5}{60}+\frac {x^3}{6}-\frac {x^2}{2}+x\right ) \]