12.13.5 problem 5

Internal problem ID [1896]
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 : 5
Date solved : Monday, January 27, 2025 at 05:38:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x +2\right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }+3 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=4\\ y^{\prime }\left (0\right )&=3 \end{align*}

✓ Solution by Maple

Time used: 0.000 (sec). Leaf size: 20

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

\[ y = 4+3 x -\frac {15}{4} x^{2}+\frac {1}{4} x^{3}+\frac {11}{16} x^{4}-\frac {5}{16} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 36

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

\[ y(x)\to -\frac {5 x^5}{16}+\frac {11 x^4}{16}+\frac {x^3}{4}-\frac {15 x^2}{4}+3 x+4 \]