12.13.6 problem 6

Internal problem ID [1897]
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 : 6
Date solved : Monday, January 27, 2025 at 05:38:01 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (x^{2}+3 x +3\right ) y^{\prime \prime }+\left (6+4 x \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 )&=7\\ y^{\prime }\left (0\right )&=3 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 20

Order:=6; 
dsolve([(3+3*x+x^2)*diff(y(x),x$2)+(6+4*x)*diff(y(x),x)+2*y(x)=0,y(0) = 7, D(y)(0) = 3],y(x),type='series',x=0);
 

\[ y = 7+3 x -\frac {16}{3} x^{2}+\frac {13}{3} x^{3}-\frac {23}{9} x^{4}+\frac {10}{9} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

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

AsymptoticDSolveValue[{(3+3*x+x^2)*D[y[x],{x,2}]+(6+4*x)*D[y[x],x]+2*y[x]==0,{y[0]==7,Derivative[1][y][0] ==3}},y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {10 x^5}{9}-\frac {23 x^4}{9}+\frac {13 x^3}{3}-\frac {16 x^2}{3}+3 x+7 \]