28.4.5 problem 5

Internal problem ID [7194]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XV. page 194
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 04:25:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 64

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

\[ y = \frac {c_1 \,x^{2} \left (1-\frac {4}{3} x +\frac {5}{6} x^{2}-\frac {1}{3} x^{3}+\frac {7}{72} x^{4}-\frac {1}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (6 x^{2}-8 x^{3}+5 x^{4}-2 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-4 x +5 x^{2}+2 x^{3}-4 x^{4}+\frac {7}{3} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 85

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

\[ y(x)\to c_1 \left (\frac {13 x^4-12 x^3-4 x^2+8 x+4}{4 x}-\frac {1}{2} x \left (5 x^2-8 x+6\right ) \log (x)\right )+c_2 \left (\frac {7 x^5}{72}-\frac {x^4}{3}+\frac {5 x^3}{6}-\frac {4 x^2}{3}+x\right ) \]

✓ Sympy. Time used: 0.306 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = C_{1} x \left (\frac {7 x^{4}}{72} - \frac {x^{3}}{3} + \frac {5 x^{2}}{6} - \frac {4 x}{3} + 1\right ) + O\left (x^{6}\right ) \]