58.2.26 problem 26

Internal problem ID [9149]
Book : Second order enumerated odes
Section : section 2
Problem number : 26
Date solved : Monday, January 27, 2025 at 05:49:46 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\left (1-\frac {1}{x}\right ) y^{\prime }+4 x^{2} y \,{\mathrm e}^{-2 x}&=4 \left (x^{3}+x^{2}\right ) {\mathrm e}^{-3 x} \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 39

dsolve(diff(y(x),x$2)+(1-1/x)*diff(y(x),x)+4*x^2*y(x)*exp(-2*x)=4*(x^2+x^3)*exp(-3*x),y(x), singsol=all)
 

\[ y = \sin \left (2 \left (x +1\right ) {\mathrm e}^{-x}\right ) c_{2} +\cos \left (2 \left (x +1\right ) {\mathrm e}^{-x}\right ) c_{1} +x \,{\mathrm e}^{-x}+{\mathrm e}^{-x} \]

✓ Solution by Mathematica

Time used: 4.276 (sec). Leaf size: 142

DSolve[D[y[x],{x,2}]+(1-1/x)*D[y[x],x]+4*x^2*y[x]*Exp[-2*x]==4*(x^2+x^3)*Exp[-3*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \cos \left (2 e^{-x} \left (\log \left (e^x\right )+1\right )\right ) \int _1^{e^x}\frac {2 \log (K[1]) (\log (K[1])+1) \sin \left (\frac {2 (\log (K[1])+1)}{K[1]}\right )}{K[1]^3}dK[1]-\sin \left (2 e^{-x} \left (\log \left (e^x\right )+1\right )\right ) \int _1^{e^x}\frac {2 \cos \left (\frac {2 (\log (K[2])+1)}{K[2]}\right ) \log (K[2]) (\log (K[2])+1)}{K[2]^3}dK[2]+c_1 \cos \left (2 e^{-x} \left (\log \left (e^x\right )+1\right )\right )-c_2 \sin \left (2 e^{-x} \left (\log \left (e^x\right )+1\right )\right ) \]