12.13.47 problem 46

Internal problem ID [1938]
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 : 46
Date solved : Monday, January 27, 2025 at 05:38:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+4 x +3\right ) y^{\prime \prime }-\left (-x^{2}+4 x +5\right ) y^{\prime }-\left (x +2\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: 0.001 (sec). Leaf size: 20

Order:=6; 
dsolve([(3+4*x+x^2)*diff(y(x),x$2)-(5+4*x-x^2)*diff(y(x),x)-(2+x)*y(x)=0,y(-2) = 2, D(y)(-2) = -1],y(x),type='series',x=-2);
 

\[ y = 2-\left (x +2\right )-\frac {7}{2} \left (x +2\right )^{2}-\frac {43}{6} \left (x +2\right )^{3}-\frac {203}{24} \left (x +2\right )^{4}-\frac {167}{30} \left (x +2\right )^{5}+\operatorname {O}\left (\left (x +2\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 43

AsymptoticDSolveValue[{(3+4*x+x^2)*D[y[x],{x,2}]-(5+4*x-x^2)*D[y[x],x]-(2+x)*y[x]==0,{y[-2]==2,Derivative[1][y][-2]==-1}},y[x],{x,-2,"6"-1}]
 

\[ y(x)\to -\frac {167}{30} (x+2)^5-\frac {203}{24} (x+2)^4-\frac {43}{6} (x+2)^3-\frac {7}{2} (x+2)^2-x \]