12.13.24 problem 27

Internal problem ID [1915]
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 : 27
Date solved : Monday, January 27, 2025 at 05:38:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (7+x \right ) y^{\prime \prime }+\left (8+2 x \right ) y^{\prime }+\left (5+x \right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} -4 \end{align*}

With initial conditions

\begin{align*} y \left (-4\right )&=1\\ y^{\prime }\left (-4\right )&=2 \end{align*}

✓ Solution by Maple

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

Order:=6; 
dsolve([(7+x)*diff(y(x),x$2)+(8+2*x)*diff(y(x),x)+(5+x)*y(x)=0,y(-4) = 1, D(y)(-4) = 2],y(x),type='series',x=-4);
 

\[ y = 1+2 \left (x +4\right )-\frac {1}{6} \left (x +4\right )^{2}-\frac {10}{27} \left (x +4\right )^{3}+\frac {19}{648} \left (x +4\right )^{4}+\frac {13}{324} \left (x +4\right )^{5}+\operatorname {O}\left (\left (x +4\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 46

AsymptoticDSolveValue[{(7+x)*D[y[x],{x,2}]+(8+2*x)*D[y[x],x]+(5+x)*y[x]==0,{y[-4]==1,Derivative[1][y][-4 ]==2}},y[x],{x,-4,"6"-1}]
 

\[ y(x)\to \frac {13}{324} (x+4)^5+\frac {19}{648} (x+4)^4-\frac {10}{27} (x+4)^3-\frac {1}{6} (x+4)^2+2 (x+4)+1 \]