Internal problem ID [9793]
Internal file name [OUTPUT/8736_Monday_June_06_2022_05_21_56_AM_70109841/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1467.
ODE order: 3.
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
[[_3rd_order, _missing_x]]
\[ \boxed {y^{\prime \prime \prime }+\operatorname {a2} y^{\prime \prime }+\operatorname {a1} y^{\prime }+\operatorname {a0} y=0} \] The characteristic equation is \[ \operatorname {a2} \,\lambda ^{2}+\lambda ^{3}+\operatorname {a1} \lambda +\operatorname {a0} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \frac {\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {\operatorname {a1}}{3}-\frac {\operatorname {a2}^{2}}{9}\right )}{\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}-\frac {\operatorname {a2}}{3}\\ \lambda _2 &= -\frac {\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}{12}+\frac {\operatorname {a1} -\frac {\operatorname {a2}^{2}}{3}}{\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}-\frac {\operatorname {a2}}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 \operatorname {a1} -\frac {2 \operatorname {a2}^{2}}{3}}{\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= -\frac {\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}{12}+\frac {\operatorname {a1} -\frac {\operatorname {a2}^{2}}{3}}{\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}-\frac {\operatorname {a2}}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 \operatorname {a1} -\frac {2 \operatorname {a2}^{2}}{3}}{\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=\text {Expression too large to display} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\left (-\frac {\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}{12}+\frac {\operatorname {a1} -\frac {\operatorname {a2}^{2}}{3}}{\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}-\frac {\operatorname {a2}}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 \operatorname {a1} -\frac {2 \operatorname {a2}^{2}}{3}}{\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}\\ y_2 &= {\mathrm e}^{\left (\frac {\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {\operatorname {a1}}{3}-\frac {\operatorname {a2}^{2}}{9}\right )}{\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}-\frac {\operatorname {a2}}{3}\right ) x}\\ y_3 &= {\mathrm e}^{\left (-\frac {\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}{12}+\frac {\operatorname {a1} -\frac {\operatorname {a2}^{2}}{3}}{\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}-\frac {\operatorname {a2}}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2 \operatorname {a1} -\frac {2 \operatorname {a2}^{2}}{3}}{\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \text {Expression too large to display} \\ \end{align*}
Verification of solutions
\[ \text {Expression too large to display} \] Verified OK.
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 590
dsolve(diff(diff(diff(y(x),x),x),x)+a2*diff(diff(y(x),x),x)+a1*diff(y(x),x)+a0*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x \left (\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {2}{3}}+\frac {\operatorname {a2} \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}{3}+\left (\operatorname {a1} -\frac {\operatorname {a2}^{2}}{3}\right ) \left (i \sqrt {3}-1\right )\right )}{\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}}+c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, \operatorname {a2}^{2}+12 i \sqrt {3}\, \operatorname {a1} -\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {2}{3}}-4 \operatorname {a2} \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}-4 \operatorname {a2}^{2}+12 \operatorname {a1} \right ) x}{12 \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}}+c_{3} {\mathrm e}^{\frac {\left (\left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {2}{3}}-2 \operatorname {a2} \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}+4 \operatorname {a2}^{2}-12 \operatorname {a1} \right ) x}{6 \left (36 \operatorname {a1} \operatorname {a2} -108 \operatorname {a0} -8 \operatorname {a2}^{3}+12 \sqrt {12 \operatorname {a0} \,\operatorname {a2}^{3}-3 \operatorname {a1}^{2} \operatorname {a2}^{2}-54 \operatorname {a1} \operatorname {a2} \operatorname {a0} +12 \operatorname {a1}^{3}+81 \operatorname {a0}^{2}}\right )^{\frac {1}{3}}}} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 84
DSolve[a0*y[x] + a1*y'[x] + a2*y''[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 e^{x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,1\right ]}+c_2 e^{x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,2\right ]}+c_3 e^{x \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2 \text {a2}+\text {$\#$1} \text {a1}+\text {a0}\&,3\right ]} \]