6.14.23 problem 23

Internal problem ID [1964]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 23
Date solved : Tuesday, September 30, 2025 at 05:21:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 47

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

\[ y = \frac {c_1 \left (1-x +\frac {1}{6} x^{2}-\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}-\frac {1}{113400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_2 \left (1-\frac {1}{3} x +\frac {1}{30} x^{2}-\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}-\frac {1}{1247400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 86

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

\[ y(x)\to \frac {c_1 \left (-\frac {x^5}{1247400}+\frac {x^4}{22680}-\frac {x^3}{630}+\frac {x^2}{30}-\frac {x}{3}+1\right )}{\sqrt {x}}+\frac {c_2 \left (-\frac {x^5}{113400}+\frac {x^4}{2520}-\frac {x^3}{90}+\frac {x^2}{6}-x+1\right )}{x} \]

✓ Sympy. Time used: 0.349 (sec). Leaf size: 70

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

\[ y{\left (x \right )} = \frac {C_{2} \left (- \frac {x^{5}}{1247400} + \frac {x^{4}}{22680} - \frac {x^{3}}{630} + \frac {x^{2}}{30} - \frac {x}{3} + 1\right )}{\sqrt {x}} + \frac {C_{1} \left (\frac {x^{6}}{7484400} - \frac {x^{5}}{113400} + \frac {x^{4}}{2520} - \frac {x^{3}}{90} + \frac {x^{2}}{6} - x + 1\right )}{x} + O\left (x^{6}\right ) \]