75.17.31 problem 581

Internal problem ID [17080]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Superposition principle. Exercises page 137
Problem number : 581
Date solved : Tuesday, January 28, 2025 at 09:50:29 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 42

dsolve(diff(y(x),x$2)+diff(y(x),x)+y(x)+1=sin(x)+x+x^2,y(x), singsol=all)
 

\[ y = c_{2} {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )+{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_{1} +x^{2}-\cos \left (x \right )-x -2 \]

✓ Solution by Mathematica

Time used: 0.828 (sec). Leaf size: 156

DSolve[D[y[x],{x,2}]+D[y[x],x]+y[x]+1==Sin[x]+x+x^2,y[x],x,IncludeSingularSolutions -> True]
 

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