Internal problem ID [1177]
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: 23.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }-2 x \left (x +1\right ) y^{\prime }+\left (x^{2}+2 x +2\right ) y-{\mathrm e}^{x} x^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 24
dsolve(x^2*diff(y(x),x$2)-2*x*(x+1)*diff(y(x),x)+(x^2+2*x+2)*y(x)=x^3*exp(x),y(x), singsol=all)
\[ y \relax (x ) = c_{2} {\mathrm e}^{x} x +{\mathrm e}^{x} c_{1} x^{2}+\frac {{\mathrm e}^{x} x^{3}}{2} \]
✓ Solution by Mathematica
Time used: 0.344 (sec). Leaf size: 210
DSolve[x^2*y''[x]-2*x*y'[x]+(x^2+2*x+2)*y[x]==x^3*Exp[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{i x} x \left (\text {HypergeometricU}(-i,0,-2 i x) \left (\int _1^x\frac {e^{(1-i) K[1]} \, _1F_1(1-i;2;-2 i K[1]) K[1]}{2 \, _1F_1(1-i;2;-2 i K[1]) \text {HypergeometricU}(1-i,1,-2 i K[1]) K[1]-\text {HypergeometricU}(-i,0,-2 i K[1]) \text {LaguerreL}(-1+i,-2 i K[1])}dK[1]+c_1\right )+2 i x \, _1F_1(1-i;2;-2 i x) \left (\int _1^x\frac {i e^{(1-i) K[2]} \text {HypergeometricU}(-i,0,-2 i K[2])}{4 \, _1F_1(1-i;2;-2 i K[2]) \text {HypergeometricU}(1-i,1,-2 i K[2]) K[2]-2 \text {HypergeometricU}(-i,0,-2 i K[2]) \text {LaguerreL}(-1+i,-2 i K[2])}dK[2]+c_2\right )\right ) \\ \end{align*}