40.13.8 problem 28

Internal problem ID [6762]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 18. Linear equations with variable coefficients (Equations of second order). Supplemetary problems. Page 120
Problem number : 28
Date solved : Monday, January 27, 2025 at 02:27:48 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 25

dsolve(x^2*diff(y(x),x$2)+(x-4*x^2)*diff(y(x),x)+(1-2*x+4*x^2)*y(x)=(x^2-x+1)*exp(x),y(x), singsol=all)
 

\[ y = {\mathrm e}^{2 x} x^{i} c_2 +{\mathrm e}^{2 x} x^{-i} c_1 +{\mathrm e}^{x} \]

✓ Solution by Mathematica

Time used: 0.250 (sec). Leaf size: 104

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

\[ y(x)\to \frac {1}{2} e^{2 x} x^{-i} \left (i x^{2 i} \Gamma (-i,x)-i x^{2 i} \Gamma (1-i,x)+i x^{2 i} \Gamma (2-i,x)-i c_2 x^{2 i}-i \Gamma (i,x)+i \Gamma (1+i,x)-i \Gamma (2+i,x)+2 c_1\right ) \]