52.2.5 problem 5

Internal problem ID [8256]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number : 5
Date solved : Sunday, March 30, 2025 at 12:50:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{3}+4 x \right ) y^{\prime \prime }-2 x y^{\prime }+6 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 72

Order:=8; 
ode:=(x^3+4*x)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+6*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 x \left (1-\frac {1}{2} x +\frac {1}{24} x^{2}+\frac {1}{48} x^{3}-\frac {1}{384} x^{4}-\frac {5}{2304} x^{5}+\frac {5}{21504} x^{6}+\frac {15}{50176} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\ln \left (x \right ) \left (-\frac {3}{2} x +\frac {3}{4} x^{2}-\frac {1}{16} x^{3}-\frac {1}{32} x^{4}+\frac {1}{256} x^{5}+\frac {5}{1536} x^{6}-\frac {5}{14336} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 +\left (1+\frac {1}{2} x -\frac {7}{4} x^{2}+\frac {31}{96} x^{3}+\frac {1}{24} x^{4}-\frac {67}{3072} x^{5}-\frac {43}{10240} x^{6}+\frac {43061}{18063360} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.229 (sec). Leaf size: 121

ode=(x^3+4*x)*D[y[x],{x,2}]-2*x*D[y[x],x]+6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {x \left (5 x^5+6 x^4-48 x^3-96 x^2+1152 x-2304\right ) \log (x)}{1536}+\frac {-229 x^6-790 x^5+2240 x^4+11840 x^3-76800 x^2+61440 x+30720}{30720}\right )+c_2 \left (\frac {5 x^7}{21504}-\frac {5 x^6}{2304}-\frac {x^5}{384}+\frac {x^4}{48}+\frac {x^3}{24}-\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.873 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*Derivative(y(x), x) + (x**3 + 4*x)*Derivative(y(x), (x, 2)) + 6*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

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