45.1.18 problem 30

Internal problem ID [7218]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.1.2 page 230
Problem number : 30
Date solved : Monday, January 27, 2025 at 02:48:21 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: 20

Order:=6; 
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);
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 34

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,"6"-1}]
 

\[ y(x)\to -\frac {x^5}{30}+\frac {x^4}{2}-\frac {x^3}{3}-2 x^2-x+2 \]