2.3.14 Problem 14

Internal problem ID [10146]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 14
Date solved : Thursday, November 27, 2025 at 10:21:44 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

Solve

\begin{align*} y^{\prime \prime \prime }+y^{\prime }+y&=x \\ 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: 0.115 (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 initial conditions the solution is

\begin{align*} y &= \frac {\left (\left (\left (-76-234 i \sqrt {3}-90 i \sqrt {31}\right ) \sqrt {93}+2418 i \sqrt {3}+930 i \sqrt {31}+372\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}+2976+\left (\left (57 i \sqrt {3}+9 i \sqrt {31}+26\right ) \sqrt {93}-589 i \sqrt {3}-93 i \sqrt {31}-310\right ) \left (108+12 \sqrt {93}\right )^{{2}/{3}}\right ) {\mathrm e}^{\frac {\left (-i \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}-12 i \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}-12\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}}}}{8928}+\frac {\left (\left (\left (234 i \sqrt {3}+90 i \sqrt {31}-76\right ) \sqrt {93}-2418 i \sqrt {3}-930 i \sqrt {31}+372\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}+2976+\left (\left (-57 i \sqrt {3}-9 i \sqrt {31}+26\right ) \sqrt {93}+589 i \sqrt {3}+93 i \sqrt {31}-310\right ) \left (108+12 \sqrt {93}\right )^{{2}/{3}}\right ) {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}+12 i \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}-12\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}}}}{8928}+\frac {\left (\left (152 \sqrt {93}-744\right ) \left (108+12 \sqrt {93}\right )^{{1}/{3}}+2976+\left (-52 \sqrt {93}+620\right ) \left (108+12 \sqrt {93}\right )^{{2}/{3}}\right ) {\mathrm e}^{-\frac {\left (\left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{2}/{3}}-12\right ) x}{6 \left (108+12 \sqrt {3}\, \sqrt {31}\right )^{{1}/{3}}}}}{8928}+x -1 \\ \end{align*}

✓ Maple. Time used: 0.397 (sec). Leaf size: 447

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

\begin{align*} \text {Solution too large to show}\end{align*}

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
 

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 1546

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

Too large to display

✗ Sympy

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