Internal problem ID [2229]
Internal file name [OUTPUT/2229_Monday_February_26_2024_09_18_33_AM_72525973/index.tex]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 23, page 106
Problem number: 30.
ODE order: 4.
ODE degree: 1.
The type(s) of ODE detected by this program : "higher_order_linear_constant_coefficients_ODE"
Maple gives the following as the ode type
[[_high_order, _missing_y]]
\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y^{\prime }=x^{3}-\frac {\cos \left (2 x \right )}{2}} \] 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 \[ y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y^{\prime } = 0 \] The characteristic equation is \[ \lambda ^{4}+2 \lambda ^{2}+\lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= -\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\\ \lambda _3 &= \frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _4 &= \frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right ) x} c_{3} +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} \\ y_3 &= {\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right ) x} \\ y_4 &= {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y^{\prime } = x^{3}-\frac {\cos \left (2 x \right )}{2} \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ x^{3}-\frac {\cos \left (2 x \right )}{2} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{\cos \left (2 x \right ), \sin \left (2 x \right )\}, \{1, x, x^{2}, x^{3}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \left \{1, {\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right ) x}, {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}, {\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}\right \} \] Since \(1\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{\cos \left (2 x \right ), \sin \left (2 x \right )\}, \{x, x^{2}, x^{3}, x^{4}\}] \] 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. \[ y_p = A_{1} \cos \left (2 x \right )+A_{2} \sin \left (2 x \right )+A_{3} x +A_{4} x^{2}+A_{5} x^{3}+A_{6} x^{4} \] The unknowns \(\{A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}\}\) 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 \[ 8 A_{1} \cos \left (2 x \right )+8 A_{2} \sin \left (2 x \right )+24 A_{6}+4 A_{4}+12 A_{5} x +24 A_{6} x^{2}-2 A_{1} \sin \left (2 x \right )+2 A_{2} \cos \left (2 x \right )+A_{3}+2 A_{4} x +3 A_{5} x^{2}+4 A_{6} x^{3} = x^{3}-\frac {\cos \left (2 x \right )}{2} \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = -{\frac {1}{17}}, A_{2} = -{\frac {1}{68}}, A_{3} = -54, A_{4} = 12, A_{5} = -2, A_{6} = {\frac {1}{4}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = -\frac {\cos \left (2 x \right )}{17}-\frac {\sin \left (2 x \right )}{68}-54 x +12 x^{2}-2 x^{3}+\frac {x^{4}}{4} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}+\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right ) x} c_{3} +{\mathrm e}^{\left (\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{12}-\frac {2}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}{6}-\frac {4}{\left (108+12 \sqrt {177}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{4}\right ) + \left (-\frac {\cos \left (2 x \right )}{17}-\frac {\sin \left (2 x \right )}{68}-54 x +12 x^{2}-2 x^{3}+\frac {x^{4}}{4}\right ) \\ \end{align*} Which simplifies to \[ y = c_{1} +{\mathrm e}^{\frac {\left (i \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}} \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}+24 i \sqrt {3}-24\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}}} c_{2} +{\mathrm e}^{-\frac {\left (\left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}-24\right ) x}{6 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}}} c_{3} +{\mathrm e}^{-\frac {x \left (\left (i \sqrt {3}-1\right ) \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}+24 i \sqrt {3}+24\right )}{12 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}}} c_{4} -\frac {\cos \left (2 x \right )}{17}-\frac {\sin \left (2 x \right )}{68}-54 x +12 x^{2}-2 x^{3}+\frac {x^{4}}{4} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} +{\mathrm e}^{\frac {\left (i \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}} \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}+24 i \sqrt {3}-24\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}}} c_{2} +{\mathrm e}^{-\frac {\left (\left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}-24\right ) x}{6 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}}} c_{3} +{\mathrm e}^{-\frac {x \left (\left (i \sqrt {3}-1\right ) \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}+24 i \sqrt {3}+24\right )}{12 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}}} c_{4} -\frac {\cos \left (2 x \right )}{17}-\frac {\sin \left (2 x \right )}{68}-54 x +12 x^{2}-2 x^{3}+\frac {x^{4}}{4} \\ \end{align*}
Verification of solutions
\[ y = c_{1} +{\mathrm e}^{\frac {\left (i \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}} \sqrt {3}+\left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}+24 i \sqrt {3}-24\right ) x}{12 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}}} c_{2} +{\mathrm e}^{-\frac {\left (\left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}-24\right ) x}{6 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}}} c_{3} +{\mathrm e}^{-\frac {x \left (\left (i \sqrt {3}-1\right ) \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {2}{3}}+24 i \sqrt {3}+24\right )}{12 \left (108+12 \sqrt {3}\, \sqrt {59}\right )^{\frac {1}{3}}}} c_{4} -\frac {\cos \left (2 x \right )}{17}-\frac {\sin \left (2 x \right )}{68}-54 x +12 x^{2}-2 x^{3}+\frac {x^{4}}{4} \] 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: 4; linear nonhomogeneous with symmetry [0,1] -> Calling odsolve with the ODE`, diff(diff(diff(_b(_a), _a), _a), _a) = _a^3-(1/2)*cos(2*_a)-_b(_a)-2*(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: 4; linear nonhomogeneous with symmetry [0,1] successful`
✓ Solution by Maple
Time used: 0.531 (sec). Leaf size: 1302
dsolve(diff(y(x),x$4)+2*diff(y(x),x$2)+diff(y(x),x)=x^3-1/2*cos(2*x),y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 5.87 (sec). Leaf size: 1293
DSolve[y''''[x]+2*y''[x]+y'[x]==x^3-1/2*Cos[2*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{x \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]} c_1}{\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]}+c_4+\frac {i x \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]\right ) \left (4+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^2\right ) \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^3 x^3+4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^2 x^2+12 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ] x+24\right )}{4 \sqrt {59} \left (-1+4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^2\right )}+\frac {i x \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^3 x^3+3 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ] \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^2 x^3+3 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^2 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ] x^3+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^3 x^3-4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^2 x^2-8 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ] \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ] x^2-4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^2 x^2+12 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ] x+12 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ] x-24\right )}{4 \sqrt {59} \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right )^4}-\frac {i \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]\right ) \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^5 \sin (2 x)}{4 \sqrt {59} \left (-1+4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^2\right )}-\frac {i \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right ) \sin (2 x)}{4 \sqrt {59} \left (-2 i+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right ) \left (2 i+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right )}-\frac {i \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^5 \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right ) \sin (2 x)}{4 \sqrt {59} \left (1-4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^2\right )}-\frac {i \cos (2 x) \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]\right ) \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^4}{2 \sqrt {59} \left (-1+4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]^2\right )}+\frac {i \cos (2 x) \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right )}{2 \sqrt {59} \left (-2 i+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right ) \left (2 i+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right )}+\frac {e^{x \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]} c_3}{\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]}+\frac {i x \left (4+\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^2\right ) \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^3 x^3+4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^2 x^2+12 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ] x+24\right ) \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right )}{4 \sqrt {59} \left (1-4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^2\right )}-\frac {i \cos (2 x) \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^4 \left (\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,3\right ]\right )}{2 \sqrt {59} \left (1-4 \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]^2\right )}+\frac {e^{x \text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]} c_2}{\text {Root}\left [\text {$\#$1}^3+2 \text {$\#$1}+1\&,2\right ]} \]