12.10.33 problem 33

Internal problem ID [1837]
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 : 33
Date solved : Monday, January 27, 2025 at 05:36:55 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} \left (x -1\right )^{2} y^{\prime \prime }+4 x y^{\prime }+2 y&=2 x \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.086 (sec). Leaf size: 67

dsolve([(x-1)^2*diff(y(x),x$2)+4*x*diff(y(x),x)+2*y(x)=2*x,y(0) = 0, D(y)(0) = -2],y(x), singsol=all)
 

\[ y = \frac {x^{3}+28 \,{\mathrm e}^{\frac {4}{x -1}} \operatorname {Ei}_{1}\left (\frac {4}{x -1}\right )-7 \,{\mathrm e}^{\frac {4 x}{x -1}}-28 \,{\mathrm e}^{\frac {4}{x -1}} \operatorname {Ei}_{1}\left (-4\right )-2 x^{2}-6 x +7}{3 \left (x -1\right )^{2}} \]

✓ Solution by Mathematica

Time used: 0.205 (sec). Leaf size: 71

DSolve[{(x-1)^2*D[y[x],{x,2}]+4*x*D[y[x],x]+2*y[x]==2*x,{y[0]==0,Derivative[1][y][0] ==-2}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {-28 e^{\frac {4}{x-1}} \operatorname {ExpIntegralEi}\left (-\frac {4}{x-1}\right )+28 \operatorname {ExpIntegralEi}(4) e^{\frac {4}{x-1}}+x^3-2 x^2-6 x-7 e^{\frac {4 x}{x-1}}+7}{3 (x-1)^2} \]