1. [2360071423 (20 pts)] Write the word TRUE or FALSE as appropriate. No work need be shown. No partial credit given.
(a) The equation of motion for a particular harmonic oscillator is $x(t)=\cos 2 t-\sin 2 t$. An equivalent expression is $x(t)=$ $\sqrt{2} \cos \left(2 t-\frac{\pi}{4}\right)$
(b) Any finite order, linear, homogeneous, constant coefficient differential equation that does not contain an undifferentiated term will always have a constant solution.
(c) The differential equation $x^{\prime \prime}+e^{x}=2$ describes a conservative system.
(d) The function $x(t)=e^{2 t}+e^{-3 t}$ is a possible solution to the differential equation for a harmonic oscillator governed by $m x^{\prime \prime}+$ $b x^{\prime}+k x=0$ with mass $m$, damping constant $b$ and restoring constant $k$.
(e) The solution space of the differential equation $y^{\prime \prime \prime}=0$ is span $\left\{t^{2}+t, t^{2}+2,4 t-8\right\}$.
Answer
(e) The solution space of the differential equation $y^{\prime \prime \prime}=0$ is span $\left\{t^{2}+t, t^{2}+2,4 t-8\right\}$. The general solution of the differential equation $y^{\prime \prime \prime}=0$ is $y(t) = a t^2 + b t + c$, where $a$, $b$, and $c$ are arbitrary constants. The given set of functions does not form a basis for the solution space, because it contains functions that are not of the form $a t^2 + b t + c$. So, the statement is \(\boxed{FALSE}\).
Step 1 ：(a) We can rewrite the given expression $x(t)=\cos 2 t-\sin 2 t$ using the trigonometric identity $\cos(a-b) = \cos a \cos b + \sin a \sin b$. We have $x(t) = \sqrt{2} \cos \left(2 t-\frac{\pi}{4}\right)$, which is equivalent to the given expression. So, the statement is \(\boxed{TRUE}\).
Step 2 ：(b) A finite order, linear, homogeneous, constant coefficient differential equation that does not contain an undifferentiated term will always have a constant solution. This is because the general solution of such a differential equation is a linear combination of exponential functions, and the constant function is a special case of an exponential function (with exponent 0). So, the statement is \(\boxed{TRUE}\).
Step 3 ：(c) The differential equation $x^{\prime \prime}+e^{x}=2$ does not describe a conservative system. A conservative system is described by a second order differential equation of the form $x^{\prime \prime}+f(x)=0$, where $f(x)$ is a function of $x$ only. In the given equation, $e^{x}$ is a function of $x$, but the equation also contains the constant term 2, which violates the condition for a conservative system. So, the statement is \(\boxed{FALSE}\).
Step 4 ：(d) The function $x(t)=e^{2 t}+e^{-3 t}$ is not a possible solution to the differential equation for a harmonic oscillator governed by $m x^{\prime \prime}+$ $b x^{\prime}+k x=0$ with mass $m$, damping constant $b$ and restoring constant $k$. This is because the given function is a sum of two exponential functions with different exponents, while the solutions to the harmonic oscillator equation are sinusoidal functions or exponential functions with complex exponents. So, the statement is \(\boxed{FALSE}\).
Step 5 ：(e) The solution space of the differential equation $y^{\prime \prime \prime}=0$ is span $\left\{t^{2}+t, t^{2}+2,4 t-8\right\}$. The general solution of the differential equation $y^{\prime \prime \prime}=0$ is $y(t) = a t^2 + b t + c$, where $a$, $b$, and $c$ are arbitrary constants. The given set of functions does not form a basis for the solution space, because it contains functions that are not of the form $a t^2 + b t + c$. So, the statement is \(\boxed{FALSE}\).
