67.24.5 problem 34.5 (e)

Internal problem ID [16972]
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 (e)
Date solved : Thursday, October 02, 2025 at 01:41:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \sinh \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-y \sin \left (x \right )&=0 \end{align*}

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 390

Order:=6; 
ode:=sinh(x)*diff(diff(y(x),x),x)+x^2*diff(y(x),x)-y(x)*sin(x) = 0; 
dsolve(ode,y(x),type='series',x=2);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.071 (sec). Leaf size: 744

ode=Sinh[x]*D[y[x],{x,2}]+x^2*D[y[x],x]-Sin[x]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,2,5}]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Sympy. Time used: 3.251 (sec). Leaf size: 228

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x) - y(x)*sin(x) + sinh(x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=2,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (x - \frac {\left (x - 2\right )^{4} \sin {\left (x + 2 \right )}}{3 \sinh ^{2}{\left (x + 2 \right )}} - \frac {\left (x - 2\right )^{4}}{12 \sinh {\left (x + 2 \right )}} + \frac {2 \left (x - 2\right )^{4}}{\sinh ^{2}{\left (x + 2 \right )}} - \frac {8 \left (x - 2\right )^{4}}{3 \sinh ^{3}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{3} \sin {\left (x + 2 \right )}}{6 \sinh {\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{3}}{3 \sinh {\left (x + 2 \right )}} + \frac {8 \left (x - 2\right )^{3}}{3 \sinh ^{2}{\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{2}}{\sinh {\left (x + 2 \right )}} - 2\right ) + C_{1} \left (\frac {\left (x - 2\right )^{4} \sin ^{2}{\left (x + 2 \right )}}{24 \sinh ^{2}{\left (x + 2 \right )}} - \frac {\left (x - 2\right )^{4} \sin {\left (x + 2 \right )}}{3 \sinh ^{2}{\left (x + 2 \right )}} + \frac {2 \left (x - 2\right )^{4} \sin {\left (x + 2 \right )}}{3 \sinh ^{3}{\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{3} \sin {\left (x + 2 \right )}}{3 \sinh ^{2}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{2} \sin {\left (x + 2 \right )}}{2 \sinh {\left (x + 2 \right )}} + 1\right ) + O\left (x^{6}\right ) \]