6.13.50 problem 49

Internal problem ID [1941]
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 : 49
Date solved : Tuesday, September 30, 2025 at 05:21:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}-11 x +16\right ) y^{\prime \prime }+\left (x^{2}-6 x +10\right ) y^{\prime }-\left (2-x \right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 3 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 20

Order:=6; 
ode:=(2*x^2-11*x+16)*diff(diff(y(x),x),x)+(x^2-6*x+10)*diff(y(x),x)-(2-x)*y(x) = 0; 
ic:=[y(3) = 1, D(y)(3) = -2]; 
dsolve([ode,op(ic)],y(x),type='series',x=3);
 

\[ y = 1-2 \left (x -3\right )+\frac {1}{2} \left (x -3\right )^{2}-\frac {1}{6} \left (x -3\right )^{3}+\frac {1}{4} \left (x -3\right )^{4}-\frac {1}{6} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 46

ode=(16-11*x+2*x^2)*D[y[x],{x,2}]+(10-6*x+x^2)*D[y[x],x]-(2-x)*y[x]==0; 
ic={y[3]==1,Derivative[1][y][3 ]==-2}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,3,5}]
 

\[ y(x)\to -\frac {1}{6} (x-3)^5+\frac {1}{4} (x-3)^4-\frac {1}{6} (x-3)^3+\frac {1}{2} (x-3)^2-2 (x-3)+1 \]

✓ Sympy. Time used: 0.399 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 2)*y(x) + (x**2 - 6*x + 10)*Derivative(y(x), x) + (2*x**2 - 11*x + 16)*Derivative(y(x), (x, 2)),0) 
ics = {y(3): 1, Subs(Derivative(y(x), x), x, 3): -2} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=3,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (x - \frac {\left (x - 3\right )^{4}}{12} + \frac {\left (x - 3\right )^{3}}{6} - \frac {\left (x - 3\right )^{2}}{2} - 3\right ) + C_{1} \left (\frac {\left (x - 3\right )^{4}}{12} + \frac {\left (x - 3\right )^{3}}{6} - \frac {\left (x - 3\right )^{2}}{2} + 1\right ) + O\left (x^{6}\right ) \]