problem 4

Internal problem ID [14046]
Internal ﬁle name [OUTPUT/13727_Friday_March_01_2024_09_08_49_AM_78086821/index.tex]

Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section: Chapter 1. Introduction to Diﬀerential Equations. Exercises 1.1, page 10
Problem number: 4.
ODE order: 3.
ODE degree: 1.

The type(s) of ODE detected by this program : "higher_order_linear_constant_coeﬃcients_ODE"

\[ \boxed {y^{\prime \prime \prime }-2 y^{\prime \prime }+5 y^{\prime }+y={\mathrm e}^{x}} \] 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 \[ y^{\prime \prime \prime }-2 y^{\prime \prime }+5 y^{\prime }+y = 0 \] The characteristic equation is \[ \lambda ^{3}-2 \lambda ^{2}+5 \lambda +1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}+\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}\\ \lambda _2 &= \frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )}{2}\\ \lambda _3 &= \frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )}{2} \end {align*}

Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )}{2}\right ) x} c_{1} +{\mathrm e}^{\left (\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}+\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}\right ) x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{\left (\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )}{2}\right ) x} \\ y_2 &= {\mathrm e}^{\left (\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )}{2}\right ) x} \\ y_3 &= {\mathrm e}^{\left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}+\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}\right ) x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }-2 y^{\prime \prime }+5 y^{\prime }+y = {\mathrm e}^{x} \] The particular solution is found using the method of undetermined coeﬃcients. Looking at the RHS of the ode, which is \[ {\mathrm e}^{x} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{{\mathrm e}^{x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \left \{{\mathrm e}^{\left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}+\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}\right ) x}, {\mathrm e}^{\left (\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )}{2}\right ) x}, {\mathrm e}^{\left (\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )}{2}\right ) x}\right \} \] 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. \[ y_p = A_{1} {\mathrm e}^{x} \] 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 \[ 5 A_{1} {\mathrm e}^{x} = {\mathrm e}^{x} \] Solving for the unknowns by comparing coeﬃcients results in \[ \left [A_{1} = {\frac {1}{5}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {{\mathrm e}^{x}}{5} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{\left (\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )}{2}\right ) x} c_{1} +{\mathrm e}^{\left (\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}+\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}\right ) x} c_{3}\right ) + \left (\frac {{\mathrm e}^{x}}{5}\right ) \\ \end{align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )}{2}\right ) x} c_{1} +{\mathrm e}^{\left (\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}+\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}\right ) x} c_{3} +\frac {{\mathrm e}^{x}}{5} \\ \end{align*}

Veriﬁcation of solutions

\[ y = {\mathrm e}^{\left (\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )}{2}\right ) x} c_{1} +{\mathrm e}^{\left (\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {\left (404+60 \sqrt {69}\right )^{{1}/{3}}}{6}+\frac {22}{3 \left (404+60 \sqrt {69}\right )^{{1}/{3}}}+\frac {2}{3}\right ) x} c_{3} +\frac {{\mathrm e}^{x}}{5} \] 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`

\[ y \left (x \right ) = \frac {{\mathrm e}^{x}}{5}+c_{1} {\mathrm e}^{\frac {\left (15 \sqrt {69}\, \left (404+60 \sqrt {69}\right )^{\frac {2}{3}}-101 \left (404+60 \sqrt {69}\right )^{\frac {2}{3}}-484 \left (404+60 \sqrt {69}\right )^{\frac {1}{3}}+1936\right ) x}{2904}}+c_{2} {\mathrm e}^{-\frac {\left (15 \sqrt {69}\, \left (404+60 \sqrt {69}\right )^{\frac {2}{3}}-101 \left (404+60 \sqrt {69}\right )^{\frac {2}{3}}-484 \left (404+60 \sqrt {69}\right )^{\frac {1}{3}}-3872\right ) x}{5808}} \cos \left (\frac {\left (404+60 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}} \sqrt {3}\, \left (15 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {23}-101 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}+484\right ) x}{5808}\right )+c_{3} {\mathrm e}^{-\frac {\left (15 \sqrt {69}\, \left (404+60 \sqrt {69}\right )^{\frac {2}{3}}-101 \left (404+60 \sqrt {69}\right )^{\frac {2}{3}}-484 \left (404+60 \sqrt {69}\right )^{\frac {1}{3}}-3872\right ) x}{5808}} \sin \left (\frac {\left (404+60 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}} \sqrt {3}\, \left (15 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {23}-101 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}+484\right ) x}{5808}\right ) \]

1.3 problem 4