85.75.11 problem 1 (k)

Internal problem ID [22976]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 7. Solution of differential equations by use of series. A Exercises at page 329
Problem number : 1 (k)
Date solved : Thursday, October 02, 2025 at 09:17:08 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} -6 y x -y^{\prime }+x \left (x^{2}+2\right ) y^{\prime \prime }&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 32

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

\[ y = c_1 \,x^{{3}/{2}} \left (1+\frac {3}{8} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+3 x^{2}+\frac {3}{5} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 45

ode=x*(x^2+2)*D[y[x],{x,2}]-D[y[x],x]-6*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {3 x^4}{5}+3 x^2+1\right )+c_1 \left (-\frac {3 x^4}{128}+\frac {3 x^2}{8}+1\right ) x^{3/2} \]

✓ Sympy. Time used: 0.480 (sec). Leaf size: 63

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {3456 x^{5}}{175} - \frac {288 x^{4}}{5} - 96 x^{3} - 72 x^{2} - 12 x + 1\right ) + C_{1} x^{\frac {3}{2}} \left (\frac {32 x^{3}}{35} + \frac {72 x^{2}}{35} + \frac {12 x}{5} + 1\right ) + O\left (x^{6}\right ) \]