12.13.42 problem 41

Internal problem ID [1933]
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 : 41
Date solved : Monday, January 27, 2025 at 05:38:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

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

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

\[ y = 2+3 x -\frac {7}{2} x^{2}-\frac {5}{6} x^{3}+\frac {41}{24} x^{4}+\frac {41}{120} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

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

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

\[ y(x)\to \frac {41 x^5}{120}+\frac {41 x^4}{24}-\frac {5 x^3}{6}-\frac {7 x^2}{2}+3 x+2 \]