54.9.17 problem 17

Internal problem ID [8687]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 17
Date solved : Wednesday, March 05, 2025 at 06:12:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 41

Order:=8; 
ode:=2*x^2*diff(diff(y(x),x),x)-x*(2*x+1)*diff(y(x),x)+(3*x+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_{1} \sqrt {x}\, \left (1-2 x +\operatorname {O}\left (x^{8}\right )\right )+c_{2} x \left (1-\frac {1}{3} x -\frac {1}{30} x^{2}-\frac {1}{210} x^{3}-\frac {1}{1512} x^{4}-\frac {1}{11880} x^{5}-\frac {1}{102960} x^{6}-\frac {1}{982800} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 70

ode=2*x^2*D[y[x],{x,2}]-x*(1+2*x)*D[y[x],x]+(1+3*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 x \left (-\frac {x^7}{982800}-\frac {x^6}{102960}-\frac {x^5}{11880}-\frac {x^4}{1512}-\frac {x^3}{210}-\frac {x^2}{30}-\frac {x}{3}+1\right )+c_2 (1-2 x) \sqrt {x} \]

✓ Sympy. Time used: 0.939 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*Derivative(y(x), (x, 2)) - x*(2*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=8)
 

\[ y{\left (x \right )} = C_{2} x \left (- \frac {x^{6}}{102960} - \frac {x^{5}}{11880} - \frac {x^{4}}{1512} - \frac {x^{3}}{210} - \frac {x^{2}}{30} - \frac {x}{3} + 1\right ) + C_{1} \sqrt {x} \left (1 - 2 x\right ) + O\left (x^{8}\right ) \]