12.13.45 problem 44

Internal problem ID [1936]
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 : 44
Date solved : Monday, January 27, 2025 at 05:38:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (3 x^{2}+12 x +13\right ) y^{\prime }+\left (5+2 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 )&=-3 \end{align*}

✓ Solution by Maple

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

Order:=6; 
dsolve([diff(y(x),x$2)+(13+12*x+3*x^2)*diff(y(x),x)+(5+2*x)*y(x)=0,y(-2) = 2, D(y)(-2) = -3],y(x),type='series',x=-2);
 

\[ y = 2-3 \left (x +2\right )+\frac {1}{2} \left (x +2\right )^{2}-\frac {1}{3} \left (x +2\right )^{3}+\frac {31}{24} \left (x +2\right )^{4}-\frac {53}{120} \left (x +2\right )^{5}+\operatorname {O}\left (\left (x +2\right )^{6}\right ) \]

✓ Solution by Mathematica

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

AsymptoticDSolveValue[{D[y[x],{x,2}]+(13+12*x+3*x^2)*D[y[x],x]+(5+2*x)*y[x]==0,{y[-2]==2,Derivative[1][y][-2]==-3}},y[x],{x,-2,"6"-1}]
 

\[ y(x)\to -\frac {53}{120} (x+2)^5+\frac {31}{24} (x+2)^4-\frac {1}{3} (x+2)^3+\frac {1}{2} (x+2)^2-3 (x+2)+2 \]