6.15.36 problem 32

Internal problem ID [2034]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS II. Exercises 7.6. Page 374
Problem number : 32
Date solved : Tuesday, September 30, 2025 at 05:22:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+7 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 36

Order:=6; 
ode:=2*x^2*(x^2+2)*diff(diff(y(x),x),x)+7*x^3*diff(y(x),x)+(3*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {3}{8} x^{2}+\frac {21}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{16} x^{2}+\frac {17}{512} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \sqrt {x} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 77

ode=2*x^2*(2+x^2)*D[y[x],{x,2}]+7*x^3*D[y[x],x]+(1+3*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {21 x^4}{128}-\frac {3 x^2}{8}+1\right )+c_2 \left (\sqrt {x} \left (\frac {17 x^4}{512}-\frac {x^2}{16}\right )+\sqrt {x} \left (\frac {21 x^4}{128}-\frac {3 x^2}{8}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.429 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(7*x**3*Derivative(y(x), x) + 2*x**2*(x**2 + 2)*Derivative(y(x), (x, 2)) + (3*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \sqrt {x} \left (\frac {84035 x^{4}}{3072} - \frac {1715 x^{3}}{96} + \frac {147 x^{2}}{16} - \frac {7 x}{2} + 1\right ) + O\left (x^{6}\right ) \]