75.17.28 problem 578

Internal problem ID [17077]
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 : 578
Date solved : Tuesday, January 28, 2025 at 09:49:55 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+5 y&={\mathrm e}^{x} \left (1-2 \sin \left (x \right )^{2}\right )+10 x +1 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 30

dsolve(diff(y(x),x$2)-2*diff(y(x),x)+5*y(x)=exp(x)*(1-2*sin(x)^2)+10*x+1,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{x} \left (x +4 c_{2} \right ) \sin \left (2 x \right )}{4}+{\mathrm e}^{x} \cos \left (2 x \right ) c_{1} +2 x +1 \]

✓ Solution by Mathematica

Time used: 0.737 (sec). Leaf size: 110

DSolve[D[y[x],{x,2}]-2*D[y[x],x]+5*y[x]==Exp[x]*(1-2*Sin[x]^2)+10*x+1,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^x \left (\cos (2 x) \int _1^x-\frac {1}{2} e^{-K[2]} \left (e^{K[2]} \cos (2 K[2])+10 K[2]+1\right ) \sin (2 K[2])dK[2]+\sin (2 x) \int _1^x\frac {1}{2} e^{-K[1]} \cos (2 K[1]) \left (e^{K[1]} \cos (2 K[1])+10 K[1]+1\right )dK[1]+c_2 \cos (2 x)+c_1 \sin (2 x)\right ) \]