2.3.14 Problem 14

Internal problem ID [8872]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 14
Date solved : Friday, February 21, 2025 at 08:43:29 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

Solve

\begin{align*} y^{\prime \prime \prime }+y^{\prime }+y&=x \end{align*}

With initial conditions

\begin{align*} y^{\prime }\left (0\right )&=0\\ y \left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=1 \end{align*}

Solved as higher order constant coeff ode

Time used: 0.138 (sec)

The characteristic equation is

\[ \lambda ^{3}+\lambda +1 = 0 \]

The roots of the above equation are

\begin{align*} \lambda _1 &= -\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\\ \lambda _2 &= \frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right )}{2}\\ \lambda _3 &= \frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right )}{2} \end{align*}

Therefore the homogeneous solution is

\[ y_h(x)={\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right )}{2}\right ) x} c_1 +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right )}{2}\right ) x} c_2 +{\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right ) x} c_3 \]

The fundamental set of solutions for the homogeneous solution are the following

\begin{align*} y_1 &= {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right )}{2}\right ) x}\\ y_2 &= {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right )}{2}\right ) x}\\ y_3 &= {\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right ) x} \end{align*}

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

\[ y^{\prime \prime \prime }+y^{\prime }+y = 0 \]

Now the particular solution to the given ODE is found

\[ y^{\prime \prime \prime }+y^{\prime }+y = x \]

The particular solution is now 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

\[ \left \{{\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right ) x}, {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right )}{2}\right ) x}, {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right )}{2}\right ) x}\right \} \]

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.

\[ y_p = A_{2} x +A_{1} \]

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

\[ A_{2} x +A_{1}+A_{2} = x \]

Solving for the unknowns by comparing coefficients results in

\[ [A_{1} = -1, A_{2} = 1] \]

Substituting the above back in the above trial solution \(y_p\), gives the particular solution

\[ y_p = x -1 \]

Therefore the general solution is

\begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right )}{2}\right ) x} c_1 +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right )}{2}\right ) x} c_2 +{\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right ) x} c_3\right ) + \left (x -1\right ) \\ \end{align*}

Solving for constants of integration using given initial conditions, the solution becomes

\begin{align*} y &= {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right )}{2}\right ) x} \left (-\frac {19 \left (\sqrt {93}-\frac {93}{19}\right ) \left (1+i \sqrt {3}\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}}{2232}+\frac {1}{3}-\frac {13 \left (i \sqrt {3}-1\right ) \left (\sqrt {93}-\frac {155}{13}\right ) \left (108+12 \sqrt {93}\right )^{{2}/{3}}}{4464}\right )+\frac {10 \,{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right )}{2}\right ) x} \sqrt {3}\, \left (\left (i \sqrt {31}\, \sqrt {3}+\frac {39 i}{5}-\frac {13 \sqrt {3}}{5}-\sqrt {31}\right ) \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}-\frac {\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}} \left (19 i+\sqrt {3}\, \left (\frac {19}{3}+i \sqrt {31}\right )+\sqrt {31}\right )}{10}+\frac {124 \sqrt {3}}{5}+\frac {36 \sqrt {31}}{5}\right )}{\left (\left (\sqrt {3}\, \sqrt {31}+9\right ) \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}-\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}+12\right ) \left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}+12\right )}+\frac {{\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right ) x} \left (\left (\sqrt {93}+3\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}+6 \sqrt {93}+4 \left (108+12 \sqrt {93}\right )^{{2}/{3}}+66\right )}{\left (3 \sqrt {93}+27\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}-3 \left (108+12 \sqrt {93}\right )^{{2}/{3}}+36}+x -1 \\ \end{align*}

Maple step by step solution

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`
 

Maple dsolve solution

Solving time : 0.631 (sec)
Leaf size : 447

dsolve([diff(diff(diff(y(x),x),x),x)+diff(y(x),x)+y(x) = x,op([D(y)(0) = 0, y(0) = 0, (D@@2)(y)(0) = 1])],y(x),singsol=all)
 

\[ y = \frac {\frac {10 \,{\mathrm e}^{-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}} \left (-12+\left (-9+\sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}\right ) x}{144}} \left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {31}+\frac {3 \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}} \sqrt {31}}{5}-\frac {6 \sqrt {3}\, \sqrt {31}}{5}-\frac {39 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}}{5}-\frac {31 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}}{5}+\frac {114}{5}\right ) \cos \left (\frac {\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}} \sqrt {3}\, \left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {31}-9 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}+12\right ) x}{144}\right )}{3}-26 \,{\mathrm e}^{-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}} \left (-12+\left (-9+\sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}\right ) x}{144}} \left (\left (\sqrt {3}-\frac {5 \sqrt {31}}{13}\right ) \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}+\frac {38 \sqrt {3}}{13}-\frac {6 \sqrt {31}}{13}\right ) \sin \left (\frac {\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}} \sqrt {3}\, \left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {31}-9 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}+12\right ) x}{144}\right )+\left (-76-\frac {10 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {31}}{3}+\sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}} \sqrt {31}+4 \sqrt {3}\, \sqrt {31}+26 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}-\frac {31 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}}{3}\right ) {\mathrm e}^{\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}} \left (-12+\left (-9+\sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}\right ) x}{72}}+3 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}} \left (x -1\right ) \left (\sqrt {3}\, \sqrt {31}-\frac {31}{3}\right )}{\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {31}-31\right )} \]

✓Mathematica DSolve solution

Solving time : 0.02 (sec)
Leaf size : 1546

DSolve[{D[y[x],{x,3}]+D[y[x],x]+y[x]==x,{Derivative[1][y][1] == 0,y[0]==0,Derivative[2][y][0] ==1}},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✗Sympy solution

Solving time : 0.000 (sec)
Leaf size : 0

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x + y(x) + Derivative(y(x), x) + Derivative(y(x), (x, 3)),0) 
ics = {Subs(Derivative(y(x), x), x, 0): 0, y(0): 0, Subs(Derivative(y(x), (x, 2)), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out