Internal problem ID [5687]
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: 8.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-2 x y^{\prime }+2 y p=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 109
Order:=8; dsolve(diff(y(x),x$2)-2*x*diff(y(x),x)+2*p*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-p \,x^{2}+\frac {p \left (p -2\right ) x^{4}}{6}-\frac {p \left (p -2\right ) \left (p -4\right ) x^{6}}{90}\right ) y \relax (0)+\left (x -\frac {\left (p -1\right ) x^{3}}{3}+\frac {\left (p^{2}-4 p +3\right ) x^{5}}{30}-\frac {\left (p^{3}-9 p^{2}+23 p -15\right ) x^{7}}{630}\right ) D\relax (y )\relax (0)+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 141
AsymptoticDSolveValue[y''[x]-2*x*y'[x]+2*p*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (-\frac {1}{630} p^3 x^7+\frac {p^2 x^7}{70}+\frac {p^2 x^5}{30}-\frac {23 p x^7}{630}-\frac {2 p x^5}{15}-\frac {p x^3}{3}+\frac {x^7}{42}+\frac {x^5}{10}+\frac {x^3}{3}+x\right )+c_1 \left (-\frac {1}{90} p^3 x^6+\frac {p^2 x^6}{15}+\frac {p^2 x^4}{6}-\frac {4 p x^6}{45}-\frac {p x^4}{3}-p x^2+1\right ) \]