54.9.7 problem 7

Internal problem ID [8677]
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 : 7
Date solved : Wednesday, March 05, 2025 at 06:12:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 40

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

\[ y = c_{1} \sqrt {x}\, \left (1+\frac {2}{3} x +\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+3 x +\frac {1}{2} x^{2}-\frac {1}{30} x^{3}+\frac {1}{280} x^{4}-\frac {1}{2520} x^{5}+\frac {1}{23760} x^{6}-\frac {1}{240240} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 69

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

\[ y(x)\to c_2 \left (-\frac {x^7}{240240}+\frac {x^6}{23760}-\frac {x^5}{2520}+\frac {x^4}{280}-\frac {x^3}{30}+\frac {x^2}{2}+3 x+1\right )+c_1 \left (\frac {2 x}{3}+1\right ) \sqrt {x} \]

✓ Sympy. Time used: 0.906 (sec). Leaf size: 56

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), (x, 2)) + (2*x + 1)*Derivative(y(x), x) - 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{7}}{240240} + \frac {x^{6}}{23760} - \frac {x^{5}}{2520} + \frac {x^{4}}{280} - \frac {x^{3}}{30} + \frac {x^{2}}{2} + 3 x + 1\right ) + C_{1} \sqrt {x} \left (\frac {2 x}{3} + 1\right ) + O\left (x^{8}\right ) \]