12.12.21 problem 23

Internal problem ID [1875]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 23
Date solved : Monday, January 27, 2025 at 05:37:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (3 x^{2}-6 x +5\right ) y^{\prime \prime }+\left (x -1\right ) y^{\prime }+12 y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 20

Order:=6; 
dsolve([(5-6*x+3*x^2)*diff(y(x),x$2)+(x-1)*diff(y(x),x)+12*y(x)=0,y(1) = -1, D(y)(1) = 1],y(x),type='series',x=1);
 

\[ y = -1+\left (x -1\right )+3 \left (x -1\right )^{2}-\frac {13}{12} \left (x -1\right )^{3}-\frac {5}{2} \left (x -1\right )^{4}+\frac {143}{160} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 40

AsymptoticDSolveValue[{(5-6*x+3*x^2)*D[y[x],{x,2}]+(x-1)*D[y[x],x]+12*y[x]==0,{y[1]==-1,Derivative[1][y][1]==1}},y[x],{x,1,"6"-1}]
 

\[ y(x)\to \frac {143}{160} (x-1)^5-\frac {5}{2} (x-1)^4-\frac {13}{12} (x-1)^3+3 (x-1)^2+x-2 \]