52.1.24 problem 22

Internal problem ID [8241]
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. Section 6.2 SOLUTIONS ABOUT ORDINARY POINTS. EXERCISES 6.2. Page 246
Problem number : 22
Date solved : Monday, January 27, 2025 at 03:46:42 PM
CAS classification : [[_2nd_order, _missing_y]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 16

Order:=8; 
dsolve([(x^2+1)*diff(y(x),x$2)+2*x*diff(y(x),x)=0,y(0) = 0, D(y)(0) = 1],y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 26

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

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