54.9.8 problem 8

Internal problem ID [8678]
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 : 8
Date solved : Sunday, March 30, 2025 at 01:23:19 PM
CAS classification : [_erf]

\begin{align*} y^{\prime \prime }+2 x y^{\prime }-8 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 42

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

\[ y = \left (\frac {4}{3} x^{4}+4 x^{2}+1\right ) y \left (0\right )+\left (x +x^{3}+\frac {1}{10} x^{5}-\frac {1}{210} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 43

ode=D[y[x],{x,2}]+2*x*D[y[x],x]-8*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {4 x^4}{3}+4 x^2+1\right )+c_2 \left (-\frac {x^7}{210}+\frac {x^5}{10}+x^3+x\right ) \]

✓ Sympy. Time used: 0.757 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = C_{2} \left (\frac {4 x^{4}}{3} + 4 x^{2} + 1\right ) + C_{1} x \left (\frac {x^{4}}{10} + x^{2} + 1\right ) + O\left (x^{8}\right ) \]