12.13.19 problem 22

Internal problem ID [1910]
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 : 22
Date solved : Monday, January 27, 2025 at 05:38:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x +4\right ) y^{\prime \prime }-\left (4+2 x \right ) y^{\prime }+\left (6+x \right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} -3 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 20

Order:=6; 
dsolve([(4+x)*diff(y(x),x$2)-(4+2*x)*diff(y(x),x)+(6+x)*y(x)=0,y(-3) = 2, D(y)(-3) = -2],y(x),type='series',x=-3);
 

\[ y = 2-2 \left (x +3\right )-\left (x +3\right )^{2}+\left (x +3\right )^{3}-\frac {11}{12} \left (x +3\right )^{4}+\frac {67}{60} \left (x +3\right )^{5}+\operatorname {O}\left (\left (x +3\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 40

AsymptoticDSolveValue[{(4+x)*D[y[x],{x,2}]-(4+2*x)*D[y[x],x]+(6+x)*y[x]==0,{y[-3]==2,Derivative[1][y][-3 ]==-2}},y[x],{x,-3,"6"-1}]
 

\[ y(x)\to \frac {67}{60} (x+3)^5-\frac {11}{12} (x+3)^4+(x+3)^3-(x+3)^2-2 (x+3)+2 \]