3.14 problem 14
Internal
problem
ID
[7852]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
3.0
Problem
number
:
14
Date
solved
:
Monday, October 21, 2024 at 04:26:59 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*}
3.14.1 Solved as higher order constant coeff ode
Time used: 0.137 (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
\[ y_h(x)={\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right )}{2}\right ) x} c_1 +{\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right ) x} c_2 +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{12}-\frac {1}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {93}\right )^{{1}/{3}}}{6}-\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right )}{2}\right ) x} c_3 \]
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}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right ) x}\\ y_3 &= {\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} \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}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\right ) x} c_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} 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 (i \sqrt {3}-1\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}}{2232}+\frac {1}{3}+\frac {13 \left (\sqrt {93}-\frac {155}{13}\right ) \left (1+i \sqrt {3}\right ) \left (108+12 \sqrt {93}\right )^{{2}/{3}}}{4464}\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}+{\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 (\sqrt {93}-\frac {155}{13}\right ) \left (i \sqrt {3}-1\right ) \left (108+12 \sqrt {93}\right )^{{2}/{3}}}{4464}\right )+x -1 \\
\end{align*}
3.14.2 Maple step by step solution
3.14.3 Maple trace
Methods for third order ODEs:
3.14.4 Maple dsolve solution
Solving time : 0.435 (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 (-12+\left (\sqrt {93}-9\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}} x}{144}} \left (\sqrt {31}\, \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}+\frac {3 \sqrt {3}\, \sqrt {31}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}}{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 {\sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}} \left (\sqrt {31}\, \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}-9 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}+12\right ) x}{144}\right )}{3}-26 \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 ) {\mathrm e}^{-\frac {\left (-12+\left (\sqrt {93}-9\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}} x}{144}} \sin \left (\frac {\sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}} \left (\sqrt {31}\, \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}-9 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}+12\right ) x}{144}\right )+\left (-76-\frac {10 \sqrt {31}\, \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}}{3}+\sqrt {3}\, \sqrt {31}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}+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 (-12+\left (\sqrt {93}-9\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}} x}{72}}+3 \left (\sqrt {3}\, \sqrt {31}-\frac {31}{3}\right ) \left (x -1\right ) \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}}{\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {31}-31\right )}
\]
3.14.5 Mathematica DSolve solution
Solving time : 0.021 (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