74.17.8 problem 8

Internal problem ID [16460]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number : 8
Date solved : Thursday, March 13, 2025 at 08:14:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 7 x y^{\prime \prime }+10 y^{\prime }+\left (-x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.054 (sec). Leaf size: 44

Order:=6; 
ode:=7*x*diff(diff(y(x),x),x)+10*diff(y(x),x)+(-x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1-\frac {1}{4} x +\frac {1}{88} x^{2}+\frac {29}{1584} x^{3}-\frac {17}{6336} x^{4}+\frac {89}{1013760} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{3}/{7}}}+c_{2} \left (1-\frac {1}{10} x +\frac {1}{340} x^{2}+\frac {113}{8160} x^{3}-\frac {929}{1011840} x^{4}+\frac {781}{38449920} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 85

ode=7*x*D[y[x],{x,2}]+10*D[y[x],x]+(1-x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {781 x^5}{38449920}-\frac {929 x^4}{1011840}+\frac {113 x^3}{8160}+\frac {x^2}{340}-\frac {x}{10}+1\right )+\frac {c_2 \left (\frac {89 x^5}{1013760}-\frac {17 x^4}{6336}+\frac {29 x^3}{1584}+\frac {x^2}{88}-\frac {x}{4}+1\right )}{x^{3/7}} \]

✓ Sympy. Time used: 0.963 (sec). Leaf size: 75

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(7*x*Derivative(y(x), (x, 2)) + (1 - x**2)*y(x) + 10*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {781 x^{5}}{38449920} - \frac {929 x^{4}}{1011840} + \frac {113 x^{3}}{8160} + \frac {x^{2}}{340} - \frac {x}{10} + 1\right ) + \frac {C_{1} \left (\frac {89 x^{5}}{1013760} - \frac {17 x^{4}}{6336} + \frac {29 x^{3}}{1584} + \frac {x^{2}}{88} - \frac {x}{4} + 1\right )}{x^{\frac {3}{7}}} + O\left (x^{6}\right ) \]