Internal problem ID [5419]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (-4 x^{2}+x \right ) y^{\prime }+\left (4 x^{2}-2 x +1\right ) y=\left (x^{2}-x +1\right ) {\mathrm e}^{x}} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 27
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)
\[ y \left (x \right ) = {\mathrm e}^{2 x} x^{i} c_{2} +{\mathrm e}^{2 x} x^{-i} c_{1} +{\mathrm e}^{x} \]
✓ Solution by Mathematica
Time used: 0.184 (sec). Leaf size: 104
DSolve[x^2*y''[x]+(x-4*x^2)*y'[x]+(1-2*x+4*x^2)*y[x]==(x^2-x+1)*Exp[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} e^{2 x} x^{-i} \left (i x^{2 i} \Gamma (-i,x)-i x^{2 i} \Gamma (1-i,x)+i x^{2 i} \Gamma (2-i,x)-i c_2 x^{2 i}-i \Gamma (i,x)+i \Gamma (1+i,x)-i \Gamma (2+i,x)+2 c_1\right ) \]