20.26.13 problem 5

Internal problem ID [4038]
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 : 5
Date solved : Tuesday, March 04, 2025 at 05:23:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x -\frac {3}{4}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 46

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+2*x^2*diff(y(x),x)+(x-3/4)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {4}{3} x +x^{2}-\frac {8}{15} x^{3}+\frac {2}{9} x^{4}-\frac {8}{105} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (-2+4 x^{2}-\frac {16}{3} x^{3}+4 x^{4}-\frac {32}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 77

ode=x^2*D[y[x],{x,2}]+2*x^2*D[y[x],x]+(x-3/4)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-2 x^{7/2}+\frac {8 x^{5/2}}{3}-2 x^{3/2}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {2 x^{11/2}}{9}-\frac {8 x^{9/2}}{15}+x^{7/2}-\frac {4 x^{5/2}}{3}+x^{3/2}\right ) \]

✓ Sympy. Time used: 0.953 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*Derivative(y(x), x) + x**2*Derivative(y(x), (x, 2)) + (x - 3/4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {3}{2}} \left (- \frac {8 x^{3}}{15} + x^{2} - \frac {4 x}{3} + 1\right ) + \frac {C_{1}}{\sqrt {x}} + O\left (x^{6}\right ) \]