Internal problem ID [13631]
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: 34.5 (g).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\frac {\left ({\mathrm e}^{x}+1\right ) y}{1-{\mathrm e}^{x}}=0} \] With the expansion point for the power series method at \(x = 3\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 167
Order:=6; dsolve(diff(y(x),x$2)+(1+exp(x))/(1-exp(x))*y(x)=0,y(x),type='series',x=3);
\[ y \left (x \right ) = \left (1+\frac {\left (1+{\mathrm e}^{3}\right ) \left (x -3\right )^{2}}{-2+2 \,{\mathrm e}^{3}}-\frac {{\mathrm e}^{3} \left (x -3\right )^{3}}{3 \left (-1+{\mathrm e}^{3}\right )^{2}}+\frac {\left ({\mathrm e}^{9}+3 \,{\mathrm e}^{6}+{\mathrm e}^{3}-1\right ) \left (x -3\right )^{4}}{24 \left (-1+{\mathrm e}^{3}\right )^{3}}+\frac {\left (-10 \,{\mathrm e}^{9}-8 \,{\mathrm e}^{6}+6 \,{\mathrm e}^{3}\right ) \left (x -3\right )^{5}}{120 \left (-1+{\mathrm e}^{3}\right )^{4}}\right ) y \left (3\right )+\left (x -3+\frac {\left ({\mathrm e}^{6}-1\right ) \left (x -3\right )^{3}}{6 \left (-1+{\mathrm e}^{3}\right )^{2}}+\frac {\left (-{\mathrm e}^{6}+{\mathrm e}^{3}\right ) \left (x -3\right )^{4}}{6 \left (-1+{\mathrm e}^{3}\right )^{3}}+\frac {\left ({\mathrm e}^{12}+6 \,{\mathrm e}^{9}-2 \,{\mathrm e}^{6}-6 \,{\mathrm e}^{3}+1\right ) \left (x -3\right )^{5}}{120 \left (-1+{\mathrm e}^{3}\right )^{4}}\right ) D\left (y \right )\left (3\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 275
AsymptoticDSolveValue[y''[x]+(1+Exp[x])/(1-Exp[x])*y[x]==0,y[x],{x,3,5}]
\[ y(x)\to c_1 \left (-\frac {\left (e^3+4 e^6+e^9\right ) (x-3)^5}{60 \left (e^3-1\right )^4}-\frac {e^3 \left (1+e^3\right ) (x-3)^5}{60 \left (e^3-1\right )^3}+\frac {e^3 \left (-1-e^3\right ) (x-3)^5}{20 \left (e^3-1\right )^3}-\frac {\left (-e^3-e^6\right ) (x-3)^4}{12 \left (e^3-1\right )^3}+\frac {\left (-1-e^3\right )^2 (x-3)^4}{24 \left (e^3-1\right )^2}-\frac {e^3 (x-3)^3}{3 \left (e^3-1\right )^2}+\frac {\left (1+e^3\right ) (x-3)^2}{2 \left (e^3-1\right )}+1\right )+c_2 \left (-\frac {\left (-e^3-e^6\right ) (x-3)^5}{20 \left (e^3-1\right )^3}+\frac {\left (1+e^3\right )^2 (x-3)^5}{120 \left (e^3-1\right )^2}-\frac {e^3 (x-3)^4}{6 \left (e^3-1\right )^2}+\frac {\left (1+e^3\right ) (x-3)^3}{6 \left (e^3-1\right )}+x-3\right ) \]