12.13.1 problem 1

Internal problem ID [1892]
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 : 1
Date solved : Monday, January 27, 2025 at 05:37:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

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

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

\[ y = 2-3 x -2 x^{2}+\frac {7}{2} x^{3}-\frac {55}{12} x^{4}+\frac {59}{8} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 34

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

\[ y(x)\to \frac {59 x^5}{8}-\frac {55 x^4}{12}+\frac {7 x^3}{2}-2 x^2-3 x+2 \]