Internal problem ID [12511]
Internal file name [OUTPUT/11164_Monday_October_16_2023_09_54_23_PM_48092678/index.tex]
Book: DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR
PUBLISHERS, Moscow 1969.
Section: Chapter 8. Differential equations. Exercises page 595
Problem number: 142.
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]]
\[ \boxed {y^{\prime \prime \prime }-3 a y^{\prime \prime }+3 a^{2} y^{\prime }-a^{3} y=0} \] The characteristic equation is \[ -a^{3}+3 a^{2} \lambda -3 a \,\lambda ^{2}+\lambda ^{3} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= a\\ \lambda _2 &= a\\ \lambda _3 &= a \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{a x} c_{1} +x \,{\mathrm e}^{a x} c_{2} +x^{2} {\mathrm e}^{a x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{a x}\\ y_2 &= x \,{\mathrm e}^{a x}\\ y_3 &= x^{2} {\mathrm e}^{a x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{a x} c_{1} +x \,{\mathrm e}^{a x} c_{2} +x^{2} {\mathrm e}^{a x} c_{3} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{a x} c_{1} +x \,{\mathrm e}^{a x} c_{2} +x^{2} {\mathrm e}^{a x} c_{3} \] 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: 0.0 (sec). Leaf size: 19
dsolve(diff(y(x),x$3)-3*a*diff(y(x),x$2)+3*a^2*diff(y(x),x)-a^3*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{a x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 23
DSolve[y'''[x]-3*a*y''[x]+3*a^2*y'[x]-a^3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{a x} (x (c_3 x+c_2)+c_1) \]