12.13.49 problem 48

Internal problem ID [1940]
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 : 48
Date solved : Monday, January 27, 2025 at 05:38:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 16

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

\[ y = 1+\frac {3}{2} \left (x -1\right )^{2}+\frac {1}{6} \left (x -1\right )^{3}-\frac {1}{8} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 32

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

\[ y(x)\to -\frac {1}{8} (x-1)^5+\frac {1}{6} (x-1)^3+\frac {3}{2} (x-1)^2+1 \]