Internal problem ID [9792]
Internal file name [OUTPUT/8735_Monday_June_06_2022_05_21_47_AM_76731332/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1466.
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, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime \prime }-3 a y^{\prime \prime }+3 a^{2} y^{\prime }-a^{3} y={\mathrm e}^{a x}} \] 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 \[ y^{\prime \prime \prime }-3 a y^{\prime \prime }+3 a^{2} y^{\prime }-a^{3} y = 0 \] The characteristic equation is \[ -a^{3}+3 a^{2} \lambda -3 a \,\lambda ^{2}+\lambda ^{3} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= a\\ \lambda _2 &= a\\ \lambda _3 &= a \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{a x} c_{1} +x \,{\mathrm e}^{a x} c_{2} +x^{2} {\mathrm e}^{a x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{a x} \\ y_2 &= x \,{\mathrm e}^{a x} \\ y_3 &= x^{2} {\mathrm e}^{a x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }-3 a y^{\prime \prime }+3 a^{2} y^{\prime }-a^{3} y = {\mathrm e}^{a x} \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ {\mathrm e}^{a x} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{{\mathrm e}^{a x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{x \,{\mathrm e}^{a x}, x^{2} {\mathrm e}^{a x}, {\mathrm e}^{a x}\} \] Since \({\mathrm e}^{a x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x \,{\mathrm e}^{a x}\}] \] Since \(x \,{\mathrm e}^{a x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{2} {\mathrm e}^{a x}\}] \] Since \(x^{2} {\mathrm e}^{a x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{3} {\mathrm e}^{a x}\}] \] Since there was duplication between the basis functions in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis function in the above updated UC_set. \[ y_p = A_{1} x^{3} {\mathrm e}^{a x} \] The unknowns \(\{A_{1}\}\) 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 \[ 6 A_{1} {\mathrm e}^{a x}+18 A_{1} x a \,{\mathrm e}^{a x}+9 A_{1} x^{2} a^{2} {\mathrm e}^{a x}-3 a \left (6 A_{1} x \,{\mathrm e}^{a x}+6 A_{1} x^{2} a \,{\mathrm e}^{a x}+A_{1} x^{3} a^{2} {\mathrm e}^{a x}\right )+3 a^{2} \left (3 A_{1} x^{2} {\mathrm e}^{a x}+A_{1} x^{3} a \,{\mathrm e}^{a x}\right ) = {\mathrm e}^{a x} \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = {\frac {1}{6}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {x^{3} {\mathrm e}^{a x}}{6} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{a x} c_{1} +x \,{\mathrm e}^{a x} c_{2} +x^{2} {\mathrm e}^{a x} c_{3}\right ) + \left (\frac {x^{3} {\mathrm e}^{a x}}{6}\right ) \\ \end{align*} Which simplifies to \[ y = {\mathrm e}^{a x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right )+\frac {x^{3} {\mathrm e}^{a x}}{6} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{a x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right )+\frac {x^{3} {\mathrm e}^{a x}}{6} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{a x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right )+\frac {x^{3} {\mathrm e}^{a x}}{6} \] 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] trying high order linear exact nonhomogeneous trying differential order: 3; missing the dependent variable checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 24
dsolve(diff(diff(diff(y(x),x),x),x)-3*a*diff(diff(y(x),x),x)+3*a^2*diff(y(x),x)-a^3*y(x)-exp(a*x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{a x} \left (\frac {1}{6} x^{3}+c_{1} +x c_{2} +x^{2} c_{3} \right ) \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 34
DSolve[-E^(a*x) - a^3*y[x] + 3*a^2*y'[x] - 3*a*y''[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{6} e^{a x} \left (x^3+6 c_3 x^2+6 c_2 x+6 c_1\right ) \]