2.3.14 problem 14

Internal problem ID [8298]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 14
Date solved : Sunday, November 10, 2024 at 03:37:11 AM
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: 0.146 (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

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

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}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\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}}}{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}}}{6}+\frac {2}{\left (108+12 \sqrt {93}\right )^{{1}/{3}}}\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}}}{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 &= \frac {4 \,{\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 (930+\left (10 \sqrt {93}+93\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}+\left (3 \sqrt {93}+31\right ) \left (108+12 \sqrt {93}\right )^{{2}/{3}}+96 \sqrt {93}\right )}{3 \left (\sqrt {93}\, \left (108+12 \sqrt {93}\right )^{{1}/{3}}-\left (108+12 \sqrt {93}\right )^{{2}/{3}}+9 \left (108+12 \sqrt {93}\right )^{{1}/{3}}+12\right ) \left (3 \sqrt {93}+31\right )}+\frac {38 \,{\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 (\frac {1446 i}{19}+\frac {\left (-i \sqrt {93}+7 i-31 \sqrt {3}-9 \sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{{2}/{3}}}{38}+\left (-i \sqrt {93}-9 i+\frac {93 \sqrt {3}}{19}+\frac {27 \sqrt {31}}{19}\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}+\frac {150 i \sqrt {93}}{19}+\frac {186 \sqrt {3}}{19}+\frac {54 \sqrt {31}}{19}\right )}{3 \left (-i \left (108+12 \sqrt {93}\right )^{{2}/{3}} \sqrt {3}-3 \left (108+12 \sqrt {93}\right )^{{2}/{3}}-12 i \sqrt {3}+36\right ) \left (3 \sqrt {93}+31\right )}+\frac {5 \,{\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 {93}+\frac {39 i}{5}-\frac {13 \sqrt {3}}{5}-\sqrt {31}\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}-\frac {\left (108+12 \sqrt {93}\right )^{{2}/{3}} \left (19 i+i \sqrt {93}+\frac {19 \sqrt {3}}{3}+\sqrt {31}\right )}{10}+\frac {124 \sqrt {3}}{5}+\frac {36 \sqrt {31}}{5}\right )}{36 \left (3 \sqrt {93}+31\right )}+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 : 0.892 (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}} x \left (-12+\left (-9+\sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}\right )}{144}} \left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {31}+\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 {\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 \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 (108+12 \sqrt {93}\right )^{{1}/{3}} x \left (-12+\left (-9+\sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}\right )}{144}} \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}\, \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 (108+12 \sqrt {93}\right )^{{1}/{3}} x \left (-12+\left (-9+\sqrt {93}\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}\right )}{72}}+3 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{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 (3 \sqrt {3}\, \sqrt {31}-31\right )} \]

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]
 

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