Internal problem ID [5370]
Internal file name [OUTPUT/4861_Sunday_February_04_2024_12_46_34_AM_82222025/index.tex]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 14. Linear equations with constant coefficients. Supplemetary problems. Page
92
Problem number: 13.
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_x]]
\[ \boxed {y^{\prime \prime \prime }-4 y^{\prime \prime }=5} \] 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 \[ y^{\prime \prime \prime }-4 y^{\prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{3}-4 \lambda ^{2} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 4\\ \lambda _2 &= 0\\ \lambda _3 &= 0 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{2} x +c_{1} +{\mathrm e}^{4 x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= x \\ y_3 &= {\mathrm e}^{4 x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }-4 y^{\prime \prime } = 5 \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 1 \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{1, x, {\mathrm e}^{4 x}\} \] Since \(1\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x\}] \] Since \(x\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{2}\}] \] 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. \[ y_p = A_{1} x^{2} \] 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 \[ -8 A_{1} = 5 \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = -{\frac {5}{8}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = -\frac {5 x^{2}}{8} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{2} x +c_{1} +{\mathrm e}^{4 x} c_{3}\right ) + \left (-\frac {5 x^{2}}{8}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{2} x +c_{1} +{\mathrm e}^{4 x} c_{3} -\frac {5 x^{2}}{8} \\ \end{align*}
Verification of solutions
\[ y = c_{2} x +c_{1} +{\mathrm e}^{4 x} c_{3} -\frac {5 x^{2}}{8} \] Verified OK.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable <- high order exact linear fully integrable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 21
dsolve(diff(y(x),x$3)-4*diff(y(x),x$2)=5,y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{4 x} c_{1}}{16}-\frac {5 x^{2}}{8}+c_{2} x +c_{3} \]
✓ Solution by Mathematica
Time used: 0.039 (sec). Leaf size: 30
DSolve[y'''[x]-4*y''[x]==5,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {5 x^2}{8}+c_3 x+\frac {1}{16} c_1 e^{4 x}+c_2 \]