43.1.11 problem 7.2.102

Internal problem ID [6856]
Book : Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section : Chapter 7. POWER SERIES METHODS. 7.2.1 Exercises. page 290
Problem number : 7.2.102
Date solved : Monday, January 27, 2025 at 02:31:46 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y x&=\frac {1}{1-x} \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 )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 16

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

\[ y = \frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {3}{40} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 56

AsymptoticDSolveValue[{D[y[x],{x,2}]-x*y[x]==1/(1-x),{}},y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {3 x^5}{40}+\frac {x^4}{12}+c_2 \left (\frac {x^4}{12}+x\right )+\frac {x^3}{6}+c_1 \left (\frac {x^3}{6}+1\right )+\frac {x^2}{2} \]