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]]
\[ \boxed {x^{2} y^{\prime \prime }-2 x \left (x +1\right ) y^{\prime }+\left (x^{2}+2 x +2\right ) y={\mathrm e}^{x} x^{3}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 20
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)
\[ y \left (x \right ) = \frac {{\mathrm e}^{x} x \left (2 c_{1} x +x^{2}+2 c_{2} \right )}{2} \]
✓ Solution by Mathematica
Time used: 0.867 (sec). Leaf size: 227
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]
\[ y(x)\to e^{i x} x \left (\operatorname {HypergeometricU}(-i,0,-2 i x) \int _1^x-\frac {i e^{(1-i) K[1]} L_i^{-1}(-2 i K[1])}{4 \operatorname {Hypergeometric1F1}(1-i,2,-2 i K[1]) \operatorname {HypergeometricU}(1-i,1,-2 i K[1]) K[1]-2 \operatorname {HypergeometricU}(-i,0,-2 i K[1]) \operatorname {LaguerreL}(-1+i,-2 i K[1])}dK[1]+L_i^{-1}(-2 i x) \int _1^x\frac {i e^{(1-i) K[2]} \operatorname {HypergeometricU}(-i,0,-2 i K[2])}{4 \operatorname {Hypergeometric1F1}(1-i,2,-2 i K[2]) \operatorname {HypergeometricU}(1-i,1,-2 i K[2]) K[2]-2 \operatorname {HypergeometricU}(-i,0,-2 i K[2]) \operatorname {LaguerreL}(-1+i,-2 i K[2])}dK[2]+c_1 \operatorname {HypergeometricU}(-i,0,-2 i x)+c_2 L_i^{-1}(-2 i x)\right ) \]