28.3.13 problem 15

Internal problem ID [7187]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XIV. page 177
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 04:25:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+\left (x -9\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 41

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+(x^2+x)*diff(y(x),x)+(x-9)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{3} \left (1-\frac {4}{7} x +\frac {5}{28} x^{2}-\frac {5}{126} x^{3}+\frac {1}{144} x^{4}-\frac {1}{990} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-86400+34560 x -4320 x^{2}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 60

ode=x^2*D[y[x],{x,2}]+(x+x^2)*D[y[x],x]+(x-9)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{x^3}-\frac {2}{5 x^2}+\frac {1}{20 x}\right )+c_2 \left (\frac {x^7}{144}-\frac {5 x^6}{126}+\frac {5 x^5}{28}-\frac {4 x^4}{7}+x^3\right ) \]

✓ Sympy. Time used: 0.321 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (x - 9)*y(x) + (x**2 + x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{3} \left (\frac {5 x^{2}}{28} - \frac {4 x}{7} + 1\right ) + \frac {C_{1} \left (\frac {x^{2}}{20} - \frac {2 x}{5} + 1\right )}{x^{3}} + O\left (x^{6}\right ) \]