12.13.18 problem 21

Internal problem ID [1909]
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 : 21
Date solved : Monday, January 27, 2025 at 05:38:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} -2 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 20

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

\[ y = 2-\left (x +2\right )-\frac {7}{2} \left (x +2\right )^{2}+\frac {4}{3} \left (x +2\right )^{3}-\frac {1}{24} \left (x +2\right )^{4}+\frac {1}{60} \left (x +2\right )^{5}+\operatorname {O}\left (\left (x +2\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 43

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

\[ y(x)\to \frac {1}{60} (x+2)^5-\frac {1}{24} (x+2)^4+\frac {4}{3} (x+2)^3-\frac {7}{2} (x+2)^2-x \]