Internal problem ID [13962]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 34. Power series solutions II: Generalization and theory. Additional Exercises.
page 678
Problem number: 34.7 (d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Titchmarsh]
\[ \boxed {y^{\prime \prime }+y \ln \left (x \right )=0} \] With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 39
Order:=6; dsolve(diff(y(x),x$2)+ln(x)*y(x)=0,y(x),type='series',x=1);
\[ y \left (x \right ) = \left (1-\frac {\left (-1+x \right )^{3}}{6}+\frac {\left (-1+x \right )^{4}}{24}-\frac {\left (-1+x \right )^{5}}{60}\right ) y \left (1\right )+\left (-1+x -\frac {\left (-1+x \right )^{4}}{12}+\frac {\left (-1+x \right )^{5}}{40}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 60
AsymptoticDSolveValue[y''[x]+Log[x]*y[x]==0,y[x],{x,1,5}]
\[ y(x)\to c_1 \left (-\frac {1}{60} (x-1)^5+\frac {1}{24} (x-1)^4-\frac {1}{6} (x-1)^3+1\right )+c_2 \left (\frac {1}{40} (x-1)^5-\frac {1}{12} (x-1)^4+x-1\right ) \]