Internal problem ID [11284]
Internal file name [OUTPUT/10270_Wednesday_December_21_2022_03_47_16_PM_68194073/index.tex]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter VII, Linear differential equations with constant coefficients. Article 52.
Summary. Page 117
Problem number: Ex 12.
ODE order: 4.
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
[[_high_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-3 y^{\prime \prime }+5 y^{\prime }-2 y={\mathrm e}^{3 x}} \] 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 \[ y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-3 y^{\prime \prime }+5 y^{\prime }-2 y = 0 \] The characteristic equation is \[ \lambda ^{4}-\lambda ^{3}-3 \lambda ^{2}+5 \lambda -2 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -2\\ \lambda _2 &= 1\\ \lambda _3 &= 1\\ \lambda _4 &= 1 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{-2 x}+{\mathrm e}^{x} c_{2} +x \,{\mathrm e}^{x} c_{3} +x^{2} {\mathrm e}^{x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{-2 x} \\ y_2 &= {\mathrm e}^{x} \\ y_3 &= x \,{\mathrm e}^{x} \\ y_4 &= x^{2} {\mathrm e}^{x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-3 y^{\prime \prime }+5 y^{\prime }-2 y = {\mathrm e}^{3 x} \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ {\mathrm e}^{3 x} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{{\mathrm e}^{3 x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{x \,{\mathrm e}^{x}, x^{2} {\mathrm e}^{x}, {\mathrm e}^{x}, {\mathrm e}^{-2 x}\} \] 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. \[ y_p = A_{1} {\mathrm e}^{3 x} \] 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 \[ 40 A_{1} {\mathrm e}^{3 x} = {\mathrm e}^{3 x} \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = {\frac {1}{40}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {{\mathrm e}^{3 x}}{40} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} {\mathrm e}^{-2 x}+{\mathrm e}^{x} c_{2} +x \,{\mathrm e}^{x} c_{3} +x^{2} {\mathrm e}^{x} c_{4}\right ) + \left (\frac {{\mathrm e}^{3 x}}{40}\right ) \\ \end{align*} Which simplifies to \[ y = {\mathrm e}^{-2 x} \left (\left (c_{4} x^{2}+c_{3} x +c_{2} \right ) {\mathrm e}^{3 x}+c_{1} \right )+\frac {{\mathrm e}^{3 x}}{40} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-2 x} \left (\left (c_{4} x^{2}+c_{3} x +c_{2} \right ) {\mathrm e}^{3 x}+c_{1} \right )+\frac {{\mathrm e}^{3 x}}{40} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-2 x} \left (\left (c_{4} x^{2}+c_{3} x +c_{2} \right ) {\mathrm e}^{3 x}+c_{1} \right )+\frac {{\mathrm e}^{3 x}}{40} \] Verified OK.
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 4; linear nonhomogeneous with symmetry [0,1] trying high order linear exact nonhomogeneous trying differential order: 4; missing the dependent variable checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 32
dsolve(diff(y(x),x$4)-diff(y(x),x$3)-3*diff(y(x),x$2)+5*diff(y(x),x)-2*y(x)=exp(3*x),y(x), singsol=all)
\[ y \left (x \right ) = \left (\left (c_{3} x^{2}+c_{4} x +c_{1} \right ) {\mathrm e}^{3 x}+c_{2} +\frac {{\mathrm e}^{5 x}}{40}\right ) {\mathrm e}^{-2 x} \]
✓ Solution by Mathematica
Time used: 0.091 (sec). Leaf size: 39
DSolve[y''''[x]-y'''[x]-3*y''[x]+5*y'[x]-2*y[x]==Exp[3*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{3 x}}{40}+c_1 e^{-2 x}+e^x (x (c_4 x+c_3)+c_2) \]