12.13.14 problem 14

Internal problem ID [1905]
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 : 14
Date solved : Monday, January 27, 2025 at 05:38:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x +3\right ) y^{\prime \prime }+\left (2 x +1\right ) y^{\prime }-\left (2-x \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: 18

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

\[ y = 1+\frac {3}{4} \left (x +1\right )^{2}-\frac {1}{12} \left (x +1\right )^{3}-\frac {1}{48} \left (x +1\right )^{4}-\frac {1}{120} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 41

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

\[ y(x)\to -\frac {1}{120} (x+1)^5-\frac {1}{48} (x+1)^4-\frac {1}{12} (x+1)^3+\frac {3}{4} (x+1)^2+1 \]