69.1.109 problem 156

Internal problem ID [14262]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 156
Date solved : Tuesday, January 28, 2025 at 06:23:36 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 38

dsolve(diff(y(x),x$2)-2*diff(y(x),x)+3*y(x)=exp(-x)*cos(x),y(x), singsol=all)
 

\[ y = {\mathrm e}^{x} \cos \left (\sqrt {2}\, x \right ) c_{1} +{\mathrm e}^{x} \sin \left (\sqrt {2}\, x \right ) c_{2} +\frac {5 \,{\mathrm e}^{-x} \left (\cos \left (x \right )-\frac {4 \sin \left (x \right )}{5}\right )}{41} \]

✓ Solution by Mathematica

Time used: 0.866 (sec). Leaf size: 113

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

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