40.11.14 problem 40

Internal problem ID [6748]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 16. Linear equations with constant coefficients (Short methods). Supplemetary problems. Page 107
Problem number : 40
Date solved : Monday, January 27, 2025 at 02:27:22 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+6 y&={\mathrm e}^{-2 x} \sec \left (x \right )^{2} \left (1+2 \tan \left (x \right )\right ) \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 85

dsolve(diff(y(x),x$2)+5*diff(y(x),x)+6*y(x)=exp(-2*x)*sec(x)^2*(1+2*tan(x)),y(x), singsol=all)
 

\[ y = \frac {\left (2 \sec \left (x \right )^{2}+2 c_1 +2 \tan \left (x \right )-6\right ) {\mathrm e}^{\left (-2+2 i\right ) x}+\left (\tan \left (x \right )^{2}+c_1 +\tan \left (x \right )\right ) {\mathrm e}^{\left (-2+4 i\right ) x}+2 c_2 \,{\mathrm e}^{\left (-3+2 i\right ) x}+c_2 \,{\mathrm e}^{\left (-3+4 i\right ) x}+\left (\tan \left (x \right )^{2}+c_1 +\tan \left (x \right )\right ) {\mathrm e}^{-2 x}+{\mathrm e}^{-3 x} c_2}{{\mathrm e}^{4 i x}+2 \,{\mathrm e}^{2 i x}+1} \]

✓ Solution by Mathematica

Time used: 0.121 (sec). Leaf size: 26

DSolve[D[y[x],{x,2}]+5*D[y[x],x]+6*y[x]==Exp[-2*x]*Sec[x]^2*(1+2*Tan[x]),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-3 x} \left (e^x \tan (x)+c_2 e^x+c_1\right ) \]