12.10.31 problem 31

Internal problem ID [1835]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number : 31
Date solved : Monday, January 27, 2025 at 05:36:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=-6 \end{align*}

✓ Solution by Maple

Time used: 0.022 (sec). Leaf size: 33

dsolve([(x-1)^2*diff(y(x),x$2)-2*(x-1)*diff(y(x),x)+2*y(x)=(x-1)^2,y(0) = 3, D(y)(0) = -6],y(x), singsol=all)
 

\[ y = \left (1-x \right ) \left (i \pi x -i \pi -\ln \left (x -1\right ) x +\ln \left (x -1\right )-2 x +3\right ) \]

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 30

DSolve[{(x-1)^2*D[y[x],{x,2}]-2*(x-1)*D[y[x],x]+2*y[x]==(x-1)^2,{y[0]==3,Derivative[1][y][0] ==-6}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to (x-1) (-i \pi (x-1)+2 x+(x-1) \log (x-1)-3) \]