Internal problem ID [9779]
Internal file name [OUTPUT/8722_Monday_June_06_2022_05_20_11_AM_33311337/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1453.
ODE order: 3.
ODE degree: 1.
The type(s) of ODE detected by this program : "higher_order_linear_constant_coefficients_ODE"
Maple gives the following as the ode type
[[_3rd_order, _missing_y]]
\[ \boxed {y^{\prime \prime \prime }-a^{2} y^{\prime }={\mathrm e}^{2 x a} \sin \left (x \right )^{2}} \] This is higher order nonhomogeneous ODE. Let the solution be \[ y = y_h + y_p \] Where \(y_h\) is the solution to the homogeneous ODE And \(y_p\) is a particular solution to the nonhomogeneous ODE. \(y_h\) is the solution to \[ y^{\prime \prime \prime }-a^{2} y^{\prime } = 0 \] The characteristic equation is \[ -a^{2} \lambda +\lambda ^{3} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= a\\ \lambda _3 &= -a \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} +{\mathrm e}^{x a} c_{2} +{\mathrm e}^{-x a} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= {\mathrm e}^{x a} \\ y_3 &= {\mathrm e}^{-x a} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }-a^{2} y^{\prime } = {\mathrm e}^{2 x a} \sin \left (x \right )^{2} \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ {\mathrm e}^{2 x a} \sin \left (x \right )^{2} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{{\mathrm e}^{2 x a}\}, \{{\mathrm e}^{2 x a} \cos \left (2 x \right ), {\mathrm e}^{2 x a} \sin \left (2 x \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{1, {\mathrm e}^{x a}, {\mathrm e}^{-x a}\} \] Since there is no duplication between the basis function in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis in the UC_set. \[ y_p = A_{1} {\mathrm e}^{2 x a}+A_{2} {\mathrm e}^{2 x a} \cos \left (2 x \right )+A_{3} {\mathrm e}^{2 x a} \sin \left (2 x \right ) \] The unknowns \(\{A_{1}, A_{2}, A_{3}\}\) are found by substituting the above trial solution \(y_p\) into the ODE and comparing coefficients. Substituting the trial solution into the ODE and simplifying gives \[ 8 A_{1} a^{3} {\mathrm e}^{2 x a}+8 A_{2} a^{3} {\mathrm e}^{2 x a} \cos \left (2 x \right )-24 A_{2} a^{2} {\mathrm e}^{2 x a} \sin \left (2 x \right )-24 A_{2} a \,{\mathrm e}^{2 x a} \cos \left (2 x \right )+8 A_{2} {\mathrm e}^{2 x a} \sin \left (2 x \right )+8 A_{3} a^{3} {\mathrm e}^{2 x a} \sin \left (2 x \right )+24 A_{3} a^{2} {\mathrm e}^{2 x a} \cos \left (2 x \right )-24 A_{3} a \,{\mathrm e}^{2 x a} \sin \left (2 x \right )-8 A_{3} {\mathrm e}^{2 x a} \cos \left (2 x \right )-a^{2} \left (2 A_{1} a \,{\mathrm e}^{2 x a}+2 A_{2} a \,{\mathrm e}^{2 x a} \cos \left (2 x \right )-2 A_{2} {\mathrm e}^{2 x a} \sin \left (2 x \right )+2 A_{3} a \,{\mathrm e}^{2 x a} \sin \left (2 x \right )+2 A_{3} {\mathrm e}^{2 x a} \cos \left (2 x \right )\right ) = {\mathrm e}^{2 x a} \sin \left (x \right )^{2} \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = \frac {1}{12 a^{3}}, A_{2} = \frac {-3 a^{3}+12 a}{36 a^{6}+196 a^{4}+224 a^{2}+64}, A_{3} = \frac {-11 a^{2}+4}{36 a^{6}+196 a^{4}+224 a^{2}+64}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {{\mathrm e}^{2 x a}}{12 a^{3}}+\frac {\left (-3 a^{3}+12 a \right ) {\mathrm e}^{2 x a} \cos \left (2 x \right )}{36 a^{6}+196 a^{4}+224 a^{2}+64}+\frac {\left (-11 a^{2}+4\right ) {\mathrm e}^{2 x a} \sin \left (2 x \right )}{36 a^{6}+196 a^{4}+224 a^{2}+64} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} +{\mathrm e}^{x a} c_{2} +{\mathrm e}^{-x a} c_{3}\right ) + \left (\frac {{\mathrm e}^{2 x a}}{12 a^{3}}+\frac {\left (-3 a^{3}+12 a \right ) {\mathrm e}^{2 x a} \cos \left (2 x \right )}{36 a^{6}+196 a^{4}+224 a^{2}+64}+\frac {\left (-11 a^{2}+4\right ) {\mathrm e}^{2 x a} \sin \left (2 x \right )}{36 a^{6}+196 a^{4}+224 a^{2}+64}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} +{\mathrm e}^{x a} c_{2} +{\mathrm e}^{-x a} c_{3} +\frac {{\mathrm e}^{2 x a}}{12 a^{3}}+\frac {\left (-3 a^{3}+12 a \right ) {\mathrm e}^{2 x a} \cos \left (2 x \right )}{36 a^{6}+196 a^{4}+224 a^{2}+64}+\frac {\left (-11 a^{2}+4\right ) {\mathrm e}^{2 x a} \sin \left (2 x \right )}{36 a^{6}+196 a^{4}+224 a^{2}+64} \\ \end{align*}
Verification of solutions
\[ y = c_{1} +{\mathrm e}^{x a} c_{2} +{\mathrm e}^{-x a} c_{3} +\frac {{\mathrm e}^{2 x a}}{12 a^{3}}+\frac {\left (-3 a^{3}+12 a \right ) {\mathrm e}^{2 x a} \cos \left (2 x \right )}{36 a^{6}+196 a^{4}+224 a^{2}+64}+\frac {\left (-11 a^{2}+4\right ) {\mathrm e}^{2 x a} \sin \left (2 x \right )}{36 a^{6}+196 a^{4}+224 a^{2}+64} \] Verified OK.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 3; linear nonhomogeneous with symmetry [0,1] -> Calling odsolve with the ODE`, diff(diff(_b(_a), _a), _a) = exp(2*_a*a)*sin(_a)^2+_b(_a)*a^2, _b(_a)` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] -> Try solving first the homogeneous part of the ODE checking if the LODE has constant coefficients <- constant coefficients successful <- solving first the homogeneous part of the ODE successful <- differential order: 3; linear nonhomogeneous with symmetry [0,1] successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 124
dsolve(diff(diff(diff(y(x),x),x),x)-a^2*diff(y(x),x)-exp(2*a*x)*sin(x)^2=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\left (-9 a^{6}+36 a^{4}\right ) \cos \left (2 x \right )+\left (-33 a^{5}+12 a^{3}\right ) \sin \left (2 x \right )+9 a^{6}+49 a^{4}+56 a^{2}+16\right ) {\mathrm e}^{2 a x}+108 \left (a^{2}+\frac {4}{9}\right ) a^{2} \left (a^{2}+1\right ) \left (c_{3} a +c_{1} {\mathrm e}^{a x}-c_{2} {\mathrm e}^{-a x}\right ) \left (a^{2}+4\right )}{108 \left (a^{2}+\frac {4}{9}\right ) a^{3} \left (a^{2}+1\right ) \left (a^{2}+4\right )} \]
✓ Solution by Mathematica
Time used: 6.285 (sec). Leaf size: 128
DSolve[-(E^(2*a*x)*Sin[x]^2) - a^2*y'[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-a x} \left (-9 \left (a^2-4\right ) a^4 e^{3 a x} \cos (2 x)-3 \left (11 a^2-4\right ) a^3 e^{3 a x} \sin (2 x)+\left (9 a^6+49 a^4+56 a^2+16\right ) \left (12 a^2 c_1 e^{2 a x}-12 a^2 c_2+e^{3 a x}\right )\right )}{12 a^3 \left (9 a^6+49 a^4+56 a^2+16\right )}+c_3 \]