20.26.16 problem 8

Internal problem ID [4041]
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 11.5. page 771
Problem number : 8
Date solved : Tuesday, March 04, 2025 at 05:23:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+7 x \,{\mathrm e}^{x} y^{\prime }+9 \left (1+\tan \left (x \right )\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 52

Order:=7; 
ode:=x^2*(x^2+1)*diff(diff(y(x),x),x)+7*x*exp(x)*diff(y(x),x)+9*(1+tan(x))*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+12 x +\frac {117}{8} x^{2}-\frac {67}{36} x^{3}+\frac {505}{256} x^{4}-\frac {262}{125} x^{5}+\frac {2443637}{2304000} x^{6}+\operatorname {O}\left (x^{7}\right )\right )+\left (\left (-31\right ) x -\frac {147}{2} x^{2}+\frac {37}{8} x^{3}-\frac {44803}{4608} x^{4}+\frac {5057587}{480000} x^{5}-\frac {3797765581}{622080000} x^{6}+\operatorname {O}\left (x^{7}\right )\right ) c_{2}}{x^{3}} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 143

ode=x^2*(x^2+1)*D[y[x],{x,2}]+7*x*Exp[x]*D[y[x],x]+9*(1+Tan[x])*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,6}]
 

\[ y(x)\to \frac {c_1 \left (\frac {2443637 x^6}{2304000}-\frac {262 x^5}{125}+\frac {505 x^4}{256}-\frac {67 x^3}{36}+\frac {117 x^2}{8}+12 x+1\right )}{x^3}+c_2 \left (\frac {-\frac {3797765581 x^6}{622080000}+\frac {5057587 x^5}{480000}-\frac {44803 x^4}{4608}+\frac {37 x^3}{8}-\frac {147 x^2}{2}-31 x}{x^3}+\frac {\left (\frac {2443637 x^6}{2304000}-\frac {262 x^5}{125}+\frac {505 x^4}{256}-\frac {67 x^3}{36}+\frac {117 x^2}{8}+12 x+1\right ) \log (x)}{x^3}\right ) \]

✓ Sympy. Time used: 10.724 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) + 7*x*exp(x)*Derivative(y(x), x) + (9*tan(x) + 9)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=7)
 

\[ y{\left (x \right )} = \frac {C_{1} \left (\frac {343 x^{3}}{6} + \frac {147 x^{2}}{2} + 21 x + 1\right )}{x^{3}} + O\left (x^{7}\right ) \]