52.1.22 problem 20

Internal problem ID [8239]
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 : 20
Date solved : Monday, January 27, 2025 at 03:46:40 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 24

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

\[ y = 2-x -2 x^{2}-\frac {1}{3} x^{3}+\frac {1}{2} x^{4}-\frac {1}{30} x^{5}-\frac {13}{180} x^{6}+\frac {1}{28} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 48

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

\[ y(x)\to \frac {x^7}{28}-\frac {13 x^6}{180}-\frac {x^5}{30}+\frac {x^4}{2}-\frac {x^3}{3}-2 x^2-x+2 \]