12.11.9 problem 19

Internal problem ID [1848]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number : 19
Date solved : Tuesday, March 04, 2025 at 01:45:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 1 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 40

Order:=6; 
ode:=(1+x)*diff(diff(y(x),x),x)+2*(x-1)^2*diff(y(x),x)+3*y(x) = 0; 
ic:=y(1) = a__0, D(y)(1) = a__1; 
dsolve([ode,ic],y(x),type='series',x=1);
 

\[ y = a_{0} +a_{1} \left (x -1\right )-\frac {3}{4} a_{0} \left (x -1\right )^{2}+\left (\frac {a_{0}}{8}-\frac {a_{1}}{4}\right ) \left (x -1\right )^{3}+\left (\frac {a_{0}}{16}-\frac {a_{1}}{48}\right ) \left (x -1\right )^{4}+\left (\frac {3 a_{0}}{64}+\frac {a_{1}}{40}\right ) \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 95

ode=(1+x)*D[y[x],{x,2}]+2*(x-1)^2*D[y[x],x]+3*y[x]==0; 
ic={y[1]==a0,Derivative[1][y][1]==a1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

\[ y(x)\to \frac {1}{20} (x-1)^5 \left (\frac {1}{4} \left (\frac {3 \text {a1}}{2}-\frac {3 \text {a0}}{4}\right )+\frac {9 \text {a0}}{8}+\frac {\text {a1}}{8}\right )+\frac {1}{12} (x-1)^4 \left (\frac {3 \text {a0}}{4}-\frac {\text {a1}}{4}\right )+\frac {1}{6} (x-1)^3 \left (\frac {3 \text {a0}}{4}-\frac {3 \text {a1}}{2}\right )-\frac {3}{4} \text {a0} (x-1)^2+\text {a0}+\text {a1} (x-1) \]

✓ Sympy. Time used: 0.922 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*(x - 1)**2*Derivative(y(x), x) + (x + 1)*Derivative(y(x), (x, 2)) + 3*y(x),0) 
ics = {y(1): a__0, Subs(Derivative(y(x), x), x, 1): a__1} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {\left (x - 1\right )^{4}}{16} + \frac {\left (x - 1\right )^{3}}{8} - \frac {3 \left (x - 1\right )^{2}}{4} + 1\right ) + C_{1} \left (x - \frac {\left (x - 1\right )^{4}}{48} - \frac {\left (x - 1\right )^{3}}{4} - 1\right ) + O\left (x^{6}\right ) \]