12.13.7 problem 7

Internal problem ID [1898]
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 : 7
Date solved : Monday, January 27, 2025 at 05:38:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

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

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

\[ y = 2+5 x -\frac {7}{4} x^{2}-\frac {3}{16} x^{3}+\frac {37}{192} x^{4}-\frac {7}{192} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

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

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

\[ y(x)\to -\frac {7 x^5}{192}+\frac {25 x^4}{192}+\frac {x^3}{16}-\frac {7 x^2}{4}+3 x+4 \]