Internal problem ID [15184]
Internal file name [OUTPUT/15185_Tuesday_April_23_2024_04_53_41_PM_43313790/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: 327.
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, _quadrature]]
\[ \boxed {y^{\prime \prime \prime \prime }=x} \] 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 \[ y^{\prime \prime \prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{4} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 0\\ \lambda _3 &= 0\\ \lambda _4 &= 0 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{4} x^{3}+c_{3} x^{2}+c_{2} x +c_{1} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= x \\ y_3 &= x^{2} \\ y_4 &= x^{3} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime } = x \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ x \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1, x\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{1, x, x^{2}, x^{3}\} \] Since \(1\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x, x^{2}\}] \] Since \(x\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{2}, x^{3}\}] \] Since \(x^{2}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{3}, x^{4}\}] \] Since \(x^{3}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x^{4}, x^{5}\}] \] 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. \[ y_p = A_{2} x^{5}+A_{1} x^{4} \] The unknowns \(\{A_{1}, A_{2}\}\) 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 \[ 120 x A_{2}+24 A_{1} = x \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = 0, A_{2} = {\frac {1}{120}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {x^{5}}{120} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{4} x^{3}+c_{3} x^{2}+c_{2} x +c_{1}\right ) + \left (\frac {x^{5}}{120}\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{4} x^{3}+c_{3} x^{2}+c_{2} x +c_{1} +\frac {1}{120} x^{5} \\ \end{align*}
Verification of solutions
\[ y = c_{4} x^{3}+c_{3} x^{2}+c_{2} x +c_{1} +\frac {1}{120} x^{5} \] Verified OK.
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 35
dsolve(diff(y(x),x$4)=x,y(x), singsol=all)
\[ y = \frac {x^{5}}{120}+\frac {c_{1} x^{3}}{6}+\frac {c_{2} x^{2}}{2}+\frac {\left (3 c_{1}^{2}+2 c_{3} \right ) x}{2}+c_{4} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 31
DSolve[y''''[x]==x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^5}{120}+c_4 x^3+c_3 x^2+c_2 x+c_1 \]