Internal problem ID [1528]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 9 Introduction to Linear Higher Order Equations. Section 9.3. Undetermined
Coefficients for Higher Order Equations. Page 495
Problem number: section 9.3, problem 31.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime }-y^{\prime \prime }+2 y^{\prime }-2 y-{\mathrm e}^{2 x} \left (\left (-x^{2}+5 x +27\right ) \cos \relax (x )+\left (9 x^{2}+13 x +2\right ) \sin \relax (x )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.346 (sec). Leaf size: 217
dsolve(1*diff(y(x),x$3)-1*diff(y(x),x$2)+2*diff(y(x),x)-2*y(x)=exp(2*x)*((27+5*x-x^2)*cos(1*x)+(2+13*x+9*x^2)*sin(1*x)),y(x), singsol=all)
\[ y \relax (x ) = -\frac {5 \,{\mathrm e}^{2 x} \cos \relax (x ) x^{2}}{3}+5 \,{\mathrm e}^{2 x} \cos \relax (x )+\frac {4 \,{\mathrm e}^{2 x} \sin \relax (x ) x^{2}}{3}+\frac {10 \,{\mathrm e}^{2 x} \sin \relax (x ) x}{3}+\frac {7 \,{\mathrm e}^{2 x} \sin \relax (x )}{3}+\frac {5 \,{\mathrm e}^{2 x} \cos \relax (x ) x}{3}+\left (\int \frac {\sqrt {2}\, \left (\cos \relax (x ) x^{2}-9 \sin \relax (x ) x^{2}-5 x \cos \relax (x )-13 \sin \relax (x ) x -27 \cos \relax (x )-2 \sin \relax (x )\right ) \left (\sqrt {2}\, \cos \left (\sqrt {2}\, x \right )-\sin \left (\sqrt {2}\, x \right )\right ) {\mathrm e}^{2 x}}{6}d x \right ) \cos \left (\sqrt {2}\, x \right )+\left (\int \frac {\sqrt {2}\, \left (\cos \relax (x ) x^{2}-9 \sin \relax (x ) x^{2}-5 x \cos \relax (x )-13 \sin \relax (x ) x -27 \cos \relax (x )-2 \sin \relax (x )\right ) \left (\sqrt {2}\, \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right )\right ) {\mathrm e}^{2 x}}{6}d x \right ) \sin \left (\sqrt {2}\, x \right )+c_{1} {\mathrm e}^{x}+c_{2} \cos \left (\sqrt {2}\, x \right )+c_{3} \sin \left (\sqrt {2}\, x \right ) \]
✓ Solution by Mathematica
Time used: 0.745 (sec). Leaf size: 243
DSolve[1*y'''[x]-1*y''[x]+2*y'[x]-2*y[x]==Exp[2*x]*((27+5*x-x^2)*Cos[1*x]+(2+13*x+9*x^2)*Sin[1*x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \cos \left (\sqrt {2} x\right ) \int _1^x\frac {e^{2 K[1]} (\cos (K[1]) ((K[1]-5) K[1]-27)-(K[1] (9 K[1]+13)+2) \sin (K[1])) \left (\sqrt {2} \cos \left (\sqrt {2} K[1]\right )-\sin \left (\sqrt {2} K[1]\right )\right )}{3 \sqrt {2}}dK[1]+\sin \left (\sqrt {2} x\right ) \int _1^x\frac {e^{2 K[2]} (\cos (K[2]) ((K[2]-5) K[2]-27)-(K[2] (9 K[2]+13)+2) \sin (K[2])) \left (\cos \left (\sqrt {2} K[2]\right )+\sqrt {2} \sin \left (\sqrt {2} K[2]\right )\right )}{3 \sqrt {2}}dK[2]+\frac {1}{3} e^{2 x} \left (5 \left (-x^2+x+3\right ) \cos (x)+(2 x (2 x+5)+7) \sin (x)\right )+c_3 e^x+c_1 \cos \left (\sqrt {2} x\right )+c_2 \sin \left (\sqrt {2} x\right ) \\ \end{align*}