84.30.9 problem 18.19

Internal problem ID [22298]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 18. Linear differential equations with variable coefficients. Supplementary problems
Problem number : 18.19
Date solved : Thursday, October 02, 2025 at 08:37:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{4} \left (x^{2}-4\right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }+\left (x^{2}-3 x +2\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 44

Order:=6; 
ode:=x^4*(x^2-4)*diff(diff(y(x),x),x)+(1+x)*diff(y(x),x)+(x^2-3*x+2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=2);
 

\[ y = c_1 \left (x -2\right )^{{61}/{64}} \left (1+\frac {1403}{32000} \left (x -2\right )-\frac {155537}{8257536} \left (x -2\right )^{2}+\frac {9382578453}{1061158912000} \left (x -2\right )^{3}-\frac {341749781779043}{85774791868416000} \left (x -2\right )^{4}+\frac {18697203541863552229}{11128448555302256640000} \left (x -2\right )^{5}+\operatorname {O}\left (\left (x -2\right )^{6}\right )\right )+c_2 \left (1-\frac {1}{134} \left (x -2\right )^{2}+\frac {26}{8777} \left (x -2\right )^{3}-\frac {11417}{13692120} \left (x -2\right )^{4}+\frac {439}{59903025} \left (x -2\right )^{5}+\operatorname {O}\left (\left (x -2\right )^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 100

ode=x^4*(x^2-4)*D[y[x],{x,2}]+(x+1)*D[y[x],x]+(x^2-3*x+2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,2,5}]
 

\[ y(x)\to c_1 \left (\frac {18697203541863552229 (x-2)^5}{11128448555302256640000}-\frac {341749781779043 (x-2)^4}{85774791868416000}+\frac {9382578453 (x-2)^3}{1061158912000}-\frac {155537 (x-2)^2}{8257536}+\frac {1403 (x-2)}{32000}+1\right ) (x-2)^{61/64}+c_2 \left (\frac {439 (x-2)^5}{59903025}-\frac {11417 (x-2)^4}{13692120}+\frac {26 (x-2)^3}{8777}-\frac {1}{134} (x-2)^2+1\right ) \]

✓ Sympy. Time used: 5.257 (sec). Leaf size: 1180

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*(x**2 - 4)*Derivative(y(x), (x, 2)) + (x + 1)*Derivative(y(x), x) + (x**2 - 3*x + 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=2,n=6)
 

\[ \text {Solution too large to show} \]