12.13.20 problem 23

Internal problem ID [1911]
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 : 23
Date solved : Monday, January 27, 2025 at 05:38:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 18

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

\[ y = -1+2 x +\frac {1}{3} x^{3}-\frac {5}{12} x^{4}+\frac {2}{5} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 29

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

\[ y(x)\to \frac {2 x^5}{5}-\frac {5 x^4}{12}+\frac {x^3}{3}+2 x-1 \]