Internal problem ID [11281]
Internal file name [OUTPUT/10267_Wednesday_December_21_2022_03_47_12_PM_79576549/index.tex]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter VII, Linear differential equations with constant coefficients. Article 52.
Summary. Page 117
Problem number: Ex 8.
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_y]]
\[ \boxed {y^{\prime \prime \prime }+2 y^{\prime \prime }+y^{\prime }=x^{2}-x} \] 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 \[ y^{\prime \prime \prime }+2 y^{\prime \prime }+y^{\prime } = 0 \] The characteristic equation is \[ \lambda ^{3}+2 \lambda ^{2}+\lambda = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= -1\\ \lambda _3 &= -1 \end {align*}
Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{-x}+x \,{\mathrm e}^{-x} c_{2} +c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{-x} \\ y_2 &= {\mathrm e}^{-x} x \\ y_3 &= 1 \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }+2 y^{\prime \prime }+y^{\prime } = x^{2}-x \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ x^{2}+x \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1, x, x^{2}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{1, {\mathrm e}^{-x} x, {\mathrm e}^{-x}\} \] Since \(1\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{x, x^{2}, x^{3}\}] \] 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. \[ y_p = A_{3} x^{3}+A_{2} x^{2}+A_{1} x \] The unknowns \(\{A_{1}, A_{2}, A_{3}\}\) 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 \[ 3 x^{2} A_{3}+2 x A_{2}+12 x A_{3}+A_{1}+4 A_{2}+6 A_{3} = x^{2}-x \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = 8, A_{2} = -{\frac {5}{2}}, A_{3} = {\frac {1}{3}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {1}{3} x^{3}-\frac {5}{2} x^{2}+8 x \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} {\mathrm e}^{-x}+x \,{\mathrm e}^{-x} c_{2} +c_{3}\right ) + \left (\frac {1}{3} x^{3}-\frac {5}{2} x^{2}+8 x\right ) \\ \end{align*} Which simplifies to \[ y = {\mathrm e}^{-x} \left (c_{2} x +c_{1} \right )+c_{3} +\frac {x^{3}}{3}-\frac {5 x^{2}}{2}+8 x \]
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-x} \left (c_{2} x +c_{1} \right )+c_{3} +\frac {x^{3}}{3}-\frac {5 x^{2}}{2}+8 x \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-x} \left (c_{2} x +c_{1} \right )+c_{3} +\frac {x^{3}}{3}-\frac {5 x^{2}}{2}+8 x \] Verified OK.
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] -> Calling odsolve with the ODE`, diff(diff(_b(_a), _a), _a) = _a^2-2*(diff(_b(_a), _a))-_b(_a)-_a, _b(_a)` *** Sublevel 2 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] -> Try solving first the homogeneous part of the ODE checking if the LODE has constant coefficients <- constant coefficients successful <- solving first the homogeneous part of the ODE successful <- differential order: 3; linear nonhomogeneous with symmetry [0,1] successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 35
dsolve(diff(y(x),x$3)+2*diff(y(x),x$2)+diff(y(x),x)=x^2-x,y(x), singsol=all)
\[ y \left (x \right ) = \left (-c_{1} x -c_{1} -c_{2} \right ) {\mathrm e}^{-x}+\frac {x^{3}}{3}-\frac {5 x^{2}}{2}+8 x +c_{3} \]
✓ Solution by Mathematica
Time used: 0.243 (sec). Leaf size: 39
DSolve[y'''[x]+2*y''[x]+y'[x]==x^2-x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{6} x \left (2 x^2-15 x+48\right )-e^{-x} (c_2 (x+1)+c_1)+c_3 \]