52.4.7 problem 15

Internal problem ID [8317]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. CHAPTER 6 IN REVIEW. Page 271
Problem number : 15
Date solved : Monday, January 27, 2025 at 03:48:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 24

Order:=8; 
dsolve([diff(y(x),x$2)+x*diff(y(x),x)+2*y(x)=0,y(0) = 3, D(y)(0) = -2],y(x),type='series',x=0);
 

\[ y = 3-2 x -3 x^{2}+x^{3}+x^{4}-\frac {1}{4} x^{5}-\frac {1}{5} x^{6}+\frac {1}{24} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 40

AsymptoticDSolveValue[{D[y[x],{x,2}]+x*D[y[x],x]+2*y[x]==0,{y[0]==3,Derivative[1][y][0] ==-2}},y[x],{x,0,"8"-1}]
 

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