12.14.13 problem 10

Internal problem ID [1954]
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 : 10
Date solved : Tuesday, March 04, 2025 at 01:46:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 10 x^{2} \left (2 x^{2}+x +1\right ) y^{\prime \prime }+x \left (66 x^{2}+13 x +13\right ) y^{\prime }-\left (10 x^{2}+4 x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 47

Order:=6; 
ode:=10*x^2*(2*x^2+x+1)*diff(diff(y(x),x),x)+x*(66*x^2+13*x+13)*diff(y(x),x)-(10*x^2+4*x+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1+x +\frac {14}{13} x^{2}-\frac {556}{897} x^{3}-\frac {5314}{9867} x^{4}+\frac {2092186}{2121405} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_2 \,x^{{1}/{5}} \left (1+\frac {3}{17} x -\frac {7}{153} x^{2}-\frac {547}{5661} x^{3}+\frac {26942}{266067} x^{4}+\frac {200432}{3991005} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 86

ode=10*x^2*(1+x+2*x^2)*D[y[x],{x,2}]+x*(13+13*x+66*x^2)*D[y[x],x]-(1+4*x+10*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [5]{x} \left (\frac {200432 x^5}{3991005}+\frac {26942 x^4}{266067}-\frac {547 x^3}{5661}-\frac {7 x^2}{153}+\frac {3 x}{17}+1\right )+\frac {c_2 \left (\frac {2092186 x^5}{2121405}-\frac {5314 x^4}{9867}-\frac {556 x^3}{897}+\frac {14 x^2}{13}+x+1\right )}{\sqrt {x}} \]

✓ Sympy. Time used: 1.274 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(10*x**2*(2*x**2 + x + 1)*Derivative(y(x), (x, 2)) + x*(66*x**2 + 13*x + 13)*Derivative(y(x), x) - (10*x**2 + 4*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt [5]{x} + \frac {C_{1}}{\sqrt {x}} + O\left (x^{6}\right ) \]