Internal problem ID [15185]
Internal file name [OUTPUT/15186_Tuesday_April_23_2024_04_53_42_PM_41726877/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 14. Differential equations admitting of
depression of their order. Exercises page 107
Problem number: 328.
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, _quadrature]]
\[ \boxed {y^{\prime \prime \prime }=x +\cos \left (x \right )} \] 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 \[ y^{\prime \prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{3} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 0\\ \lambda _3 &= 0 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{3} x^{2}+c_{2} x +c_{1} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= x \\ y_3 &= x^{2} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime } = x +\cos \left (x \right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ x +\cos \left (x \right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1, x\}, \{\cos \left (x \right ), \sin \left (x \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{1, x, x^{2}\} \] Since \(1\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x, x^{2}\}, \{\cos \left (x \right ), \sin \left (x \right )\}] \] Since \(x\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{2}, x^{3}\}, \{\cos \left (x \right ), \sin \left (x \right )\}] \] Since \(x^{2}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{3}, x^{4}\}, \{\cos \left (x \right ), \sin \left (x \right )\}] \] 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. \[ y_p = A_{2} x^{4}+A_{1} x^{3}+A_{3} \cos \left (x \right )+A_{4} \sin \left (x \right ) \] The unknowns \(\{A_{1}, A_{2}, A_{3}, A_{4}\}\) 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 \[ 24 A_{2} x +6 A_{1}+A_{3} \sin \left (x \right )-A_{4} \cos \left (x \right ) = x +\cos \left (x \right ) \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = 0, A_{2} = {\frac {1}{24}}, A_{3} = 0, A_{4} = -1\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {x^{4}}{24}-\sin \left (x \right ) \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{3} x^{2}+c_{2} x +c_{1}\right ) + \left (\frac {x^{4}}{24}-\sin \left (x \right )\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{3} x^{2}+c_{2} x +c_{1} +\frac {x^{4}}{24}-\sin \left (x \right ) \\ \end{align*}
Verification of solutions
\[ y = c_{3} x^{2}+c_{2} x +c_{1} +\frac {x^{4}}{24}-\sin \left (x \right ) \] Verified OK.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 24
dsolve(diff(y(x),x$3)=x+cos(x),y(x), singsol=all)
\[ y = \frac {x^{4}}{24}+\frac {c_{1} x^{2}}{2}-\sin \left (x \right )+c_{2} x +c_{3} \]
✓ Solution by Mathematica
Time used: 0.025 (sec). Leaf size: 29
DSolve[y'''[x]==x+Cos[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^4}{24}+c_3 x^2-\sin (x)+c_2 x+c_1 \]