problem 19.3 (b)

Internal problem ID [13629]
Internal ﬁle name [OUTPUT/12801_Saturday_February_17_2024_08_44_53_AM_17645592/index.tex]

Book: Ordinary Diﬀerential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section: Chapter 19. Arbitrary Homogeneous linear equations with constant coeﬃcients. Additional exercises page 369
Problem number: 19.3 (b).
ODE order: 3.
ODE degree: 1.

The type(s) of ODE detected by this program : "higher_order_linear_constant_coeﬃcients_ODE"

\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }-8 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 5, y^{\prime }\left (0\right ) = 13, y^{\prime \prime }\left (0\right ) = 86] \end {align*}

The characteristic equation is \[ \lambda ^{3}-6 \lambda ^{2}+12 \lambda -8 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 2\\ \lambda _2 &= 2\\ \lambda _3 &= 2 \end {align*}

Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} c_{2} +x^{2} {\mathrm e}^{2 x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{2 x}\\ y_2 &= x \,{\mathrm e}^{2 x}\\ y_3 &= x^{2} {\mathrm e}^{2 x} \end {align*}

Looking at the above solution \begin {align*} y = c_{1} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} c_{2} +x^{2} {\mathrm e}^{2 x} c_{3} \tag {1} \end {align*}

Initial conditions are now substituted in the above solution. This will generate the required equations to solve for the integration constants. substituting \(y = 5\) and \(x = 0\) in the above gives \begin {align*} 5 = c_{1}\tag {1A} \end {align*}

Taking derivative of the solution gives \begin {align*} y^{\prime } = 2 c_{1} {\mathrm e}^{2 x}+c_{2} {\mathrm e}^{2 x}+2 x \,{\mathrm e}^{2 x} c_{2} +2 x \,{\mathrm e}^{2 x} c_{3} +2 x^{2} {\mathrm e}^{2 x} c_{3} \end {align*}

substituting \(y^{\prime } = 13\) and \(x = 0\) in the above gives \begin {align*} 13 = 2 c_{1} +c_{2}\tag {2A} \end {align*}

Taking two derivatives of the solution gives \begin {align*} y^{\prime \prime } = 4 c_{1} {\mathrm e}^{2 x}+4 c_{2} {\mathrm e}^{2 x}+4 x \,{\mathrm e}^{2 x} c_{2} +2 \,{\mathrm e}^{2 x} c_{3} +8 x \,{\mathrm e}^{2 x} c_{3} +4 x^{2} {\mathrm e}^{2 x} c_{3} \end {align*}

substituting \(y^{\prime \prime } = 86\) and \(x = 0\) in the above gives \begin {align*} 86 = 4 c_{1} +4 c_{2} +2 c_{3}\tag {3A} \end {align*}

Equations {1A,2A,3A} are now solved for \(\{c_{1}, c_{2}, c_{3}\}\). Solving for the constants gives \begin {align*} c_{1}&=5\\ c_{2}&=3\\ c_{3}&=27 \end {align*}

Substituting these values back in above solution results in \begin {align*} y = 27 x^{2} {\mathrm e}^{2 x}+3 x \,{\mathrm e}^{2 x}+5 \,{\mathrm e}^{2 x} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{2 x} \left (27 x^{2}+3 x +5\right ) \\ \end{align*}

Veriﬁcation of solutions

\[ y = {\mathrm e}^{2 x} \left (27 x^{2}+3 x +5\right ) \] 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`

12.14 problem 19.3 (b)