Question

Deal with the initial value problem $$ u^{\prime \prime}+0.125 u^{\prime}+u=F(t), \quad u(0)=2, \quad u^{\prime}(0)=0 $$ (a) Plot the given forcing function \(F(t)\) versus \(t\) and also plot the solution \(u(t)\) versus \(t\) on the same set of axes. Use a \(t\) interval that is long enough so the initial transients are substantially eliminated. Observe the relation between the amplitude and phase of the forcing term and the amplitude and phase of the response. Note that \(\omega_{0}=\sqrt{k / m}=1\). (b) Draw the phase plot of the solution, that is, plot \(u^{\prime}\) versus \(u .\) \(F(t)=3 \cos (0.3 t)\)

Short answer

Question: Plot the forcing function \(F(t) = 3 \cos(0.3t)\) and the solution \(u(t) = e^{-0.0625 t}((-1.244)\cos(t) + 0.2558\sin(t)) + 3.244\cos(0.3t) - 0.120\sin(0.3t)\) on the same set of axes. Observe the amplitude and phase of the response with respect to the forcing term. Also, draw the phase plot of the solution by plotting \(u'\) versus \(u\).

Step-by-step solution

Find the general solution of the homogeneous equation

First, let's consider the homogeneous part of the equation: $$ u^{\prime \prime}+0.125 u^{\prime}+u=0 $$ This is a linear homogeneous second-order differential equation with constant coefficients. The characteristic equation is: $$ r^2+0.125 r +1=0 $$ This equation has complex roots: \(r = -0.0625 \pm i\), which corresponds to a solution of the form $$ u_h(t) = e^{-0.0625 t}(A\cos(t) + B\sin(t)) $$

Find the particular solution using the method of undetermined coefficients

Now we find the particular solution for the non-homogeneous part. Since \(F(t) = 3\cos(0.3t)\), we propose a particular solution in the form: $$ u_p(t) = C \cos(0.3t)+D\sin(0.3t) $$ Now we find the first and second derivatives and substitute them into the original equation: $$ u_p'(t) = -0.3C \sin(0.3t)+0.3D\cos(0.3t) $$ $$ u_p''(t) = -0.09C \cos(0.3t) - 0.09D\sin(0.3t) $$ Substituting this back into the original equation, we get: $$ (-0.09 C+0.125(0.3 D)+C)\cos(0.3t)+ (-0.09 D -0.125(0.3 C)+D)\sin(0.3t) = 3\cos(0.3t) $$ Now we equate coefficients to find C and D: $$ 0.91C+0.0375D = 3 $$ $$ 0.0375C+0.91D = 0 $$ Solving this system of equations, we get \(C \approx 3.244\) and \(D \approx -0.120\). Thus, our particular solution is: $$ u_p(t) = 3.244\cos(0.3t) - 0.120\sin(0.3t) $$

Combine the homogeneous and particular solutions

Combining the homogeneous and particular solutions, the general solution is: $$ u(t) = e^{-0.0625 t}(A\cos(t) + B\sin(t)) + 3.244\cos(0.3t) - 0.120\sin(0.3t) $$

Apply the initial conditions

Now we apply the initial conditions to find the constants A and B: $$ u(0) = 2 = A + 3.244 $$ $$ u'(0) = 0 = -0.0625A + 0.3B $$ Solving these equations, we find \(A = -1.244\) and \(B = 0.2558\). Therefore, our specific solution is: $$ u(t) = e^{-0.0625 t}((-1.244)\cos(t) + 0.2558\sin(t)) + 3.244\cos(0.3t) - 0.120\sin(0.3t) $$ Now that we have the specific solution, we can plot the given forcing function \(F(t)\) and the solution \(u(t)\) on the same set of axes and draw the phase plot of the solution.

Key concepts

Characteristic Equation

The cornerstone of solving linear homogeneous differential equations is the characteristic equation. The characteristic equation emerges from the substitution of an exponential trial solution into the homogeneous differential equation. This process transforms the differential equation into an algebraic equation.

Regarding our problem, the equation \(u'' + 0.125u' + u = 0\) leads us to the characteristic equation \(r^2 + 0.125r + 1 = 0\). Solving for \(r\) gives us complex roots, indicating the solution will involve oscillatory motion, damped by a factor of \(e^{-0.0625t}\). This complex nature will contribute to the behavior of the system's homogeneous solution over time.

Homogeneous Solutions

Homogeneous solutions, also termed the complementary or natural response, refer to the part of the solution that satisfies the homogeneous differential equation: \(u'' + 0.125u' + u = 0\).

In our discussion, the homogeneous solution \(u_h(t) = e^{-0.0625t}(A\cos(t) + B\sin(t))\) represents the system's response without any external forcing. It highlights how initial conditions influence the system independently of the input force and demonstrates the system's natural frequencies and damping characteristics. The parameters \(A\) and \(B\) will eventually be solved using the given initial conditions, revealing the nature of the transient behavior.

Particular Solutions

Opposite to the homogeneous solutions are the particular solutions. They solely account for the system's forced response due to an external stimulus or non-homogeneous part of the differential equation. In technical terms, the particular solution does not involve constants derived from initial conditions.

For the equation \(u'' + 0.125u' + u = F(t)\), where \(F(t) = 3\cos(0.3t)\), we assume a particular solution of similar form to the forcing function \(u_p(t) = C\cos(0.3t) + D\sin(0.3t)\). Adjusting the coefficients \(C\) and \(D\) through the method of undetermined coefficients, we engineer a solution that perfectly fits the external force.

Method of Undetermined Coefficients

The method of undetermined coefficients is a technique for determining the particular solution to non-homogeneous linear differential equations. To utilize this method, we propose a form for the particular solution based on the non-homogeneous term \(F(t)\), and then we adjust its coefficients to satisfy the original differential equation.

In our exercise, this method involves guessing the form \(u_p(t) = C\cos(0.3t) + D\sin(0.3t)\) and finding the expressions for its derivatives. These are substituted back into the differential equation to yield a set of algebraic equations for \(C\) and \(D\). Solving this system gives us the coefficients that will fit the non-homogeneous term, thus providing the particular solution linked to the external force.

Phase Plot

In dynamical systems, a phase plot is a graphical representation that depicts the trajectory of a system in phase space with the system's state variables along the axes. For a second-order system, the phase plot is a curve that represents the state variable \(u\) and its derivative \(u'\) against each other.

In the context of our initial value problem, the phase plot allows visualization of how the solution evolves over time without directly considering time itself. This plot, which shows \(u'\) versus \(u\), facilitates analysis of the system's stability and behavior. It is particularly useful for identifying patterns such as spirals or centers typical for systems with complex roots in the characteristic equation, as demonstrated by the trajectory formed by the solution to our differential equation.