Question

Find the general solutions of the following equations and compare computer solutions.

$\mathbf{(}{\mathbf{D}}^{\mathbf{4}}\mathbf{+}\mathbf{4}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{0}$Hint: Find the four 4th roots of -4 (see Chapter 2, Section 10).

Short answer

The general solution is,$y=(c_1{e}^{ix}+c_2{e}^{-ix})e^x+(c_3{e}^{ix}+c_4{e}^{-ix})e^{-x}$

Step-by-step solution

Given information from question

Given equation is.$(D^4+4)y=0$

Differential equation

A differential equation is a formula that connects the derivatives of one or more unknown functions. Functions are used to represent physical quantities, derivatives are used to characterise their rates of change, and differential equations are used to define a relationship between them in applications.

Calculate the general solution of (D4+4)y=0

Rewriting the auxiliary equation

$\begin{array}{c}[{\left(D^2\right)}^2-{\left(i2\right)}^2]y=(D^2-i2)(D^2+i2)y\\ =0\end{array}$

To find the general solution of this forth order differential equation. It needs to treat $(D^2+\sqrt{i2})y=0$and$(D^2+\sqrt{i2})y=0$separately. For$(D^2+i2)y=0$it has the roots$(D^2-\sqrt{i2})(D^2+\sqrt{i2})y=0$

Any complex number's root could be written as:

$\begin{array}{c}{z}^{1/2}={\left(r{e}^{i\theta }\right)}^{1/2}\\ =\sqrt{r}(\cos \frac{\theta }{n}+i\sin \frac{\theta }{n})\end{array}$

The polar coordinates for $2i$are$r=2$ and $\theta =\pi /2$therefore

$\begin{array}{c}2e^{\pi /2}=\sqrt{2}{e}^{\pi /4}\\ =\sqrt{2}(\cos \frac{\theta }{4}+i\sin \frac{\theta }{4})\\ =1+i\end{array}$

Therefore, roots of the equation become, .$(D-1-i)(D+1+i)y=0$

Therefore, the solution is

$y=(c_1{e}^{ix}+c_2{e}^{-ix})e^x$

Where for the part of ,$D^{2}y=0$ the roots would be

$\begin{array}{c}(D-i\sqrt{i2})(D+i\sqrt{i2})y=0\\ (D-i)(1+i)(D+i(1+i\left)\right)y=0\\ (D-i+1)(D+i-1)y=0\end{array}$

Therefore, the solution is,

$y(c_3{e}^{ix}+c_4{e}^{-ix})e^{ix}$

Thus, the general solution is

$y=(c_1{e}^{ix}+c_2{e}^{-ix})e^x+(c_3{e}^{ix}+c_4{e}^{-ix})e^{-x}$