6.13.4 problem 4

Internal problem ID [1895]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 05:21:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (3 x^{2}+x +1\right ) y^{\prime \prime }+\left (2+15 x \right ) y^{\prime }+12 y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 18

Order:=6; 
ode:=(3*x^2+x+1)*diff(diff(y(x),x),x)+(2+15*x)*diff(y(x),x)+12*y(x) = 0; 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = x -x^{2}-\frac {7}{2} x^{3}+\frac {15}{2} x^{4}+\frac {45}{8} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 31

ode=(1+x+3*x^2)*D[y[x],{x,2}]+(2+15*x)*D[y[x],x]+12*y[x]==0; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {45 x^5}{8}+\frac {15 x^4}{2}-\frac {7 x^3}{2}-x^2+x \]

✓ Sympy. Time used: 0.328 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((15*x + 2)*Derivative(y(x), x) + (3*x**2 + x + 1)*Derivative(y(x), (x, 2)) + 12*y(x),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (18 x^{4} + 6 x^{3} - 6 x^{2} + 1\right ) + C_{1} x \left (\frac {15 x^{3}}{2} - \frac {7 x^{2}}{2} - x + 1\right ) + O\left (x^{6}\right ) \]