problem Problem 41

Internal problem ID [12192]
Internal ﬁle name [OUTPUT/10845_Thursday_September_21_2023_05_47_57_AM_43712024/index.tex]

Book: Diﬀerential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section: Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number: Problem 41.
ODE order: 6.
ODE degree: 1.

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

\[ \boxed {y^{\left (6\right )}-3 y^{\left (5\right )}+3 y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }=x} \] 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 \[ y^{\left (6\right )}-3 y^{\left (5\right )}+3 y^{\prime \prime \prime \prime }-y^{\prime \prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{6}-3 \lambda ^{5}+3 \lambda ^{4}-\lambda ^{3} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 0\\ \lambda _3 &= 0\\ \lambda _4 &= 1\\ \lambda _5 &= 1\\ \lambda _6 &= 1 \end {align*}

Therefore the homogeneous solution is \[ y_h(x)=c_{3} x^{2}+c_{2} x +c_{1} +{\mathrm e}^{x} c_{4} +x \,{\mathrm e}^{x} c_{5} +x^{2} {\mathrm e}^{x} c_{6} \] 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 &= {\mathrm e}^{x} \\ y_5 &= x \,{\mathrm e}^{x} \\ y_6 &= x^{2} {\mathrm e}^{x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\left (6\right )}-3 y^{\left (5\right )}+3 y^{\prime \prime \prime \prime }-y^{\prime \prime \prime } = x \] 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 \[ \{1, x, x^{2}, x \,{\mathrm e}^{x}, x^{2} {\mathrm e}^{x}, {\mathrm e}^{x}\} \] 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 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. \[ y_p = A_{2} x^{4}+A_{1} x^{3} \] 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 \[ -24 x A_{2}-6 A_{1}+72 A_{2} = x \] Solving for the unknowns by comparing coeﬃcients results in \[ \left [A_{1} = -{\frac {1}{2}}, A_{2} = -{\frac {1}{24}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = -\frac {1}{24} x^{4}-\frac {1}{2} x^{3} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{3} x^{2}+c_{2} x +c_{1} +{\mathrm e}^{x} c_{4} +x \,{\mathrm e}^{x} c_{5} +x^{2} {\mathrm e}^{x} c_{6}\right ) + \left (-\frac {1}{24} x^{4}-\frac {1}{2} x^{3}\right ) \\ \end{align*} Which simpliﬁes to \[ y = \left (c_{6} x^{2}+c_{5} x +c_{4} \right ) {\mathrm e}^{x}+c_{3} x^{2}+c_{2} x +c_{1} -\frac {x^{4}}{24}-\frac {x^{3}}{2} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \left (c_{6} x^{2}+c_{5} x +c_{4} \right ) {\mathrm e}^{x}+c_{3} x^{2}+c_{2} x +c_{1} -\frac {x^{4}}{24}-\frac {x^{3}}{2} \\ \end{align*}

Veriﬁcation of solutions

\[ y = \left (c_{6} x^{2}+c_{5} x +c_{4} \right ) {\mathrm e}^{x}+c_{3} x^{2}+c_{2} x +c_{1} -\frac {x^{4}}{24}-\frac {x^{3}}{2} \] 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: 6; linear nonhomogeneous with symmetry [0,1] 
-> Calling odsolve with the ODE`, diff(diff(diff(_b(_a), _a), _a), _a) = 3*(diff(diff(_b(_a), _a), _a))-3*(diff(_b(_a), _a))+_b(_a)+ 
   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 
<- differential order: 6; linear nonhomogeneous with symmetry [0,1] successful`

\[ y \left (x \right ) = \left (c_{3} x^{2}+\left (c_{2} -6 c_{3} \right ) x +c_{1} -3 c_{2} +12 c_{3} \right ) {\mathrm e}^{x}-\frac {x^{4}}{24}-\frac {x^{3}}{2}+\frac {c_{4} x^{2}}{2}+c_{5} x +c_{6} \]

\[ y(x)\to -\frac {x^4}{24}-\frac {x^3}{2}+c_6 x^2+c_3 e^x \left (x^2-6 x+12\right )+c_5 x+c_1 e^x+c_2 e^x (x-3)+c_4 \]

2.30 problem Problem 41