20.25.8 problem 9

Internal problem ID [4013]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page 758
Problem number : 9
Date solved : Tuesday, March 04, 2025 at 05:22:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x \cos \left (x \right ) y^{\prime }+5 \,{\mathrm e}^{2 x} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 47

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-x*cos(x)*diff(y(x),x)+5*exp(2*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = c_{1} x^{1-2 i} \left (1+\left (-\frac {10}{17}-\frac {40 i}{17}\right ) x +\left (-\frac {365}{136}+\frac {13 i}{17}\right ) x^{2}+\left (\frac {223}{1020}+\frac {1723 i}{765}\right ) x^{3}+\left (\frac {114911}{78336}+\frac {24835 i}{78336}\right ) x^{4}+\left (\frac {4041077}{8029440}-\frac {1112267 i}{1605888}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{1+2 i} \left (1+\left (-\frac {10}{17}+\frac {40 i}{17}\right ) x +\left (-\frac {365}{136}-\frac {13 i}{17}\right ) x^{2}+\left (\frac {223}{1020}-\frac {1723 i}{765}\right ) x^{3}+\left (\frac {114911}{78336}-\frac {24835 i}{78336}\right ) x^{4}+\left (\frac {4041077}{8029440}+\frac {1112267 i}{1605888}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 94

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

\[ y(x)\to \left (\frac {11}{391680}+\frac {7 i}{391680}\right ) c_1 \left ((32064-31693 i) x^4-(30784+60608 i) x^3-(80352-23904 i) x^2+(23040+69120 i) x+(25344-16128 i)\right ) x^{1+2 i}+\left (\frac {7}{391680}+\frac {11 i}{391680}\right ) c_2 \left ((31693-32064 i) x^4+(60608+30784 i) x^3-(23904-80352 i) x^2-(69120+23040 i) x+(16128-25344 i)\right ) x^{1-2 i} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*cos(x)*Derivative(y(x), x) + 5*y(x)*exp(2*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : Expected Expr or iterable but got None