12.10.23 problem 23

Internal problem ID [1827]
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
Date solved : Monday, January 27, 2025 at 05:36:35 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+\left (x^{2}+2 x +2\right ) y&=x^{3} {\mathrm e}^{x} \end{align*}

✓ Solution by Maple

Time used: 0.008 (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)
 

\[ y = \frac {{\mathrm e}^{x} x \left (2 c_1 x +x^{2}+2 c_2 \right )}{2} \]

✓ Solution by Mathematica

Time used: 0.855 (sec). Leaf size: 227

DSolve[x^2*D[y[x],{x,2}]-2*x*D[y[x],x]+(x^2+2*x+2)*y[x]==x^3*Exp[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{i x} x \left (\operatorname {HypergeometricU}(-i,0,-2 i x) \int _1^x-\frac {i e^{(1-i) K[1]} L_i^{-1}(-2 i K[1])}{4 \operatorname {Hypergeometric1F1}(1-i,2,-2 i K[1]) \operatorname {HypergeometricU}(1-i,1,-2 i K[1]) K[1]-2 \operatorname {HypergeometricU}(-i,0,-2 i K[1]) \operatorname {LaguerreL}(-1+i,-2 i K[1])}dK[1]+L_i^{-1}(-2 i x) \int _1^x\frac {i e^{(1-i) K[2]} \operatorname {HypergeometricU}(-i,0,-2 i K[2])}{4 \operatorname {Hypergeometric1F1}(1-i,2,-2 i K[2]) \operatorname {HypergeometricU}(1-i,1,-2 i K[2]) K[2]-2 \operatorname {HypergeometricU}(-i,0,-2 i K[2]) \operatorname {LaguerreL}(-1+i,-2 i K[2])}dK[2]+c_1 \operatorname {HypergeometricU}(-i,0,-2 i x)+c_2 L_i^{-1}(-2 i x)\right ) \]