67.24.17 problem 34.7 (c)

Internal problem ID [16984]
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 (c)
Date solved : Thursday, October 02, 2025 at 01:41:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-3 x y^{\prime }+y \sin \left (x \right )&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 39

Order:=4; 
ode:=x*diff(diff(y(x),x),x)-3*x*diff(y(x),x)+y(x)*sin(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{2} x^{3}\right ) y \left (0\right )+\left (x +\frac {3}{2} x^{2}+\frac {4}{3} x^{3}\right ) y^{\prime }\left (0\right )+O\left (x^{4}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 42

ode=x*D[y[x],{x,2}]-3*x*D[y[x],x]+Sin[x]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,3}]
 

\[ y(x)\to c_1 \left (-\frac {x^3}{2}-\frac {x^2}{2}+1\right )+c_2 \left (\frac {4 x^3}{3}+\frac {3 x^2}{2}+x\right ) \]

✓ Sympy. Time used: 1.294 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x*Derivative(y(x), x) + x*Derivative(y(x), (x, 2)) + y(x)*sin(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=4)
 

\[ y{\left (x \right )} = C_{2} x \left (\frac {3 x^{2}}{2} + \frac {3 x}{2} + 1\right ) + C_{1} + O\left (x^{4}\right ) \]