7.15.33 problem 33

Internal problem ID [489]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 33
Date solved : Saturday, March 29, 2025 at 04:54:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (5 x^{3}+2 x^{2}\right ) y^{\prime \prime }+\left (-x^{2}+3 x \right ) y^{\prime }-\left (1+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 48

Order:=6; 
ode:=(5*x^3+2*x^2)*diff(diff(y(x),x),x)+(-x^2+3*x)*diff(y(x),x)-(1+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{3}/{2}} \left (1+\frac {11}{20} x -\frac {11}{224} x^{2}+\frac {671}{24192} x^{3}-\frac {9577}{387072} x^{4}+\frac {2806061}{100638720} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+10 x +5 x^{2}+\frac {10}{9} x^{3}-\frac {7}{18} x^{4}+\frac {13}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 84

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

\[ y(x)\to c_1 \sqrt {x} \left (\frac {2806061 x^5}{100638720}-\frac {9577 x^4}{387072}+\frac {671 x^3}{24192}-\frac {11 x^2}{224}+\frac {11 x}{20}+1\right )+\frac {c_2 \left (\frac {13 x^5}{45}-\frac {7 x^4}{18}+\frac {10 x^3}{9}+5 x^2+10 x+1\right )}{x} \]

✓ Sympy. Time used: 1.108 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-x - 1)*y(x) + (-x**2 + 3*x)*Derivative(y(x), x) + (5*x**3 + 2*x**2)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt {x} + \frac {C_{1}}{x} + O\left (x^{6}\right ) \]