Find the general solution to the following differential equations.
(Make sure to find the correct roots to your characteristic polynomial!!!)
\(y'' + 3y' + 2y = 0\)
\(x'' + 3x' = 0\)
\(4y'' - y = 0\)
\(y'' + 2y' + 2y = 0\)
\(4y'' + 4y' + 5y = 0\)
\(\ddot{z} + 9z = 0\)
\(y'' - 6y' + 9y = 0\)
\(4y'' + 12y' + 9y = 0\)
\(5y'' - 4y' + \frac{4}{5}y = 0\)
Find the solution to the following initial value problems. As \(t \rightarrow \infty\), what happens to the solution of the following problems? (e.g. Does it approach 0? \(-\infty\)? \(\infty\)? a certain value?)
\(2y'' + y' - 3y = 0 \hspace{1in} y(0) = 0, \hspace{.4in} y'(0) = 1\)
\(6\ddot{w} - 5\dot{w} + w = 0 \hspace{1in} w(0) = 4, \hspace{.4in} \dot{w}(0) = 0\)
\(y'' + 4y' + 3y = 0 \hspace{1in} y(0) = 2, \hspace{.4in} y'(0) = -1\)
\(w'' + 4w' + 5w = 0 \hspace{1in} w(0) = 1, \hspace{.4in} w'(0) = 0\)
\(y'' - 2y' + 5y = 0 \,, \hspace{1in} y(\frac{\pi}{2}) = 0 \,, \hspace{.4in} y'(\frac{\pi}{2}) = 2 \,.\)
\(y'' + 4y = 0 \hspace{1.4in} y(0) = 0, \hspace{.45in} y'(0) = 1\)
\(y'' + 4y' + 4y = 0 \hspace{1in} y(-1) = 2, \hspace{.3in} y'(-1) = 1\)
\(4z'' + 12z' + 9z = 0 \hspace{0.8in} z(0) = 1, \hspace{.4in} z'(0) = -4\)
\(9y'' - 12y' + 4y = 0 \hspace{0.8in} y(0) = 2, \hspace{.4in} y'(0) = -1\)
For #19 and #20, which values of \(\alpha\) do ALL possible solutions tend to \(0\) as \(t \rightarrow \infty\)?
\(y'' + 2(\alpha - 1)y' + \alpha(\alpha - 2)y = 0\)
\(y'' + (5 - \alpha)y' - 3(\alpha - 2)y = 0\)
Find a general solution of \(\Dop^4z + 18\Dop^2z + 81z = 0\), where \(\Dop = \frac{d}{dw}\).
Given the roots of the characteristic polynomial to be \(-1, 0, 2, 2, 2, -3 \pm i, -3 \pm i\), give a general solution as well as the order of the original differential equation.
Solve the initial-value problem: \[w'' -w' - 2w =0 \ , w(0) = \alpha \ , w'(0) = 2 \ .\] Find \(\alpha\) so that the solution approaches \(0\) as \(z\) approaches \(+\infty\).
Solve the initial-value problem:
\[4y'' - y = 0 \ , y(0) = 2 \ , y'(0) = \beta.\] Then find \(\beta\) so that the solution approaches \(0\) as \(x\) approaches \(+\infty\).
The following two problems constitute an exploration of \(\bf{Euler's~formula}.\)
(a) Use Euler’s formula to write \(e^{3 + 5i}\) in the form \(a + ib\) .
(b) Use Euler’s formula to write \(e^{2 - (\frac{\pi}{2})i}\) in the form \(a + ib\) .
(c) Use Euler’s formula to write \(3^{1 - i}\) in the form \(a + ib\) .
(a) Show that \(x_1(t) = \cos(t)\) and \(x_2(t) = \sin(t)\) are a fundamental set of solutions of \(x'' + x = 0\) ; that is, show their are solutions of the differential equation and that their Wronskian is not zero.
(b) Show that \(x = e^{it}\) is also a solution of \(x'' + x = 0\) . Therefore, this implies \(e^{it} = c_1\cos(t) + c_2\sin(t)\) , for some constants \(c_1\) and \(c_2\). Can you explain why?
(c) Set \(t = 0\) in the equation above. What do you obtain?
(d) Given that \(\frac{d}{dt}(e^{it}) = ie^{it}\), differentiate \(e^{it} = c_1\cos(t) + c_2\sin(t)\) and set \(t = 0\) in this newly obtained equation. What formula do you now obtain? (\(\bf{Hint}\) Given the preamble above, you should end up with Euler’s Formula. Now you’ve proved it! :))
Solve the initial value problem \(y'' - 6y' + 9\alpha y = 0\), \(y(0) = 0\). Describe and study its behavior for increasing \(t\) (i.e. take t approaching \(+\infty\)) and varrying values of the real parameter \(\alpha\). Is your solution to the second-order linear homogeneous equation above unique? Why? Why not?
\(\bf{Hint:}\) In your exploration of \(\alpha\), your argument should contain three cases.
Consider the initial-value problem:
\[4y'' + 12y' + 9y = 0 \ , y(0) = 1 \ , y'(0) = -4 \ .\]
(a) Solve the initial-value problem;
(b) Determine where the solutions attains the value \(0\) ;
(c) Determine the coordinates of the minimum point (if there is one) \((x^*, y^*)\);
(d) Change the second initial condition to \(y'(0) = \beta\) and rewrite your solution in terms of \(\beta\). Next, find the value of \(\beta\) that separates solutions that are always negative from solutions that are always positive (\(\bf{Hint:}\) Not a trick question. Such a value of \(\beta\) does indeed exist :) ).
Given the second-order homogeneous ordinary differential equation \(w'' + (2+\alpha)w' + 2\alpha w = 0\), determine what the general solution to the differential equation looks like given varrying values of the real parameter \(\alpha\).
Consider the initial value problem
\[w'' + 5 w' + 6 w = 0,\ w(0) = 2,\ w'(0) =3.\] Find the maximum value attained by the solution.
Write down a second-order, linear homogeneous differential equation with constant coefficients that has solutions:
\(X_1(t) = e^t\) and \(X_2(t) = e^{-4t};\)
\(Z_1(w) = e^{3w}\) and \(Z_2(w) = e^{2w};\)
\(Y_1(x) = e^{2x}\) and \(Y_2(x) = x \cdot e^{2x};\)
\(Y_1(t) = e^{\frac{5}{4} t}\) and \(Y_2(t) = te^{\frac{5}{4} t};\)
\(Z_1(x) = e^x \cos(2x)\) and \(Z_2(x) = e^x \sin(2x).\)
\(W_1(t) = e^t\) and \(W_2(t) = e^{\alpha t}\), where \(\alpha \neq 1\), a real parameter.
\(W_1(u) = e^{\alpha u}\cos(\beta u)\) and \(W_2(u) = e^{\alpha u} \sin(\beta u)\), where \(\alpha\) and \(\beta\) are real parameters.
On page 5 of the Lecture Notes, work out the details in the computation of the Wronskian
\( det \left (\begin{array}{ccc} e^{rt} & r e^{rt}& r^2 e^{rt} \\ t e^{rt} & rt e^{rt} + e^ {rt} & r^2 t e^{rt} + 2r e^{rt} \\ t^2 e^{rt} & r t^2 e^{rt} + 2t e^{rt} & 2e^{rt} + 4t e^{rt} + r^2 t^2 e^{rt} \end{array} \right) \), of the solutions \(Y_1(t) = e^{rt}, Y_2(t) = t e^{rt}\) and \(Y_3(t) = t^2 e^{rt}.\)
What can you conclude from here?
Suppose that \(Y_1(x)\) and \(Y_2(x)\) are solutions of the differential equation
\[\ddot{y} + 2\dot{y} + (1 + x^2)y = 0.\] Suppose you know that \(W[Y_1,Y_2](0) = 5 . \) What is \(W[Y_1,Y_2](x)? \)
Suppose that \(X_1(t)\) and \(X_2(t)\) are solutions of the differential equation
\[x''' + 2cos(t)x' + (e^t + t^2)x = 0.\] Suppose you know that \(W[X_1,X_2](1) = 3578 . \) What is \(W[Y_1,Y_2](25)? \)
Compute the Wronskian of the functions \(X_1(u) = \cos(5u)\) and \(X_2(u) = \sin (5u).\) (Evaluate the determinant and simplify)
For each of the following determine if the statement is TRUE or FALSE and justify your response:
(a) The functions \(f(w) = w^2 + 3\) and \(g(w) = w^2 + 2\) (defined on the real line) are linearly independent.
(b) Let \(y\) be the function defined on the real line by the rule \( y(u) = 1 - \cos(u)\). This function cannot be the solution of a differential equation of the form \[\ddot{y} + p(u)\dot{y} + q(u)y = 0\] where \(p\) and \(q\) are continuous on the real line.
\(\bf{Remark}\) In answering this question, you might have to cite material (or review :) ) material from previous chapters.
Consider the following statement:
“If \(y = f(x)\) and \(y= g(x)\) are solutions to the differential equation \(y'' + 4y' + 2y = e^x\) then \(|f(x) - g(x)|\) goes to zero as t goes to \(+\infty\).” Is this statement true or false? Briefly justify your answer.
\(\bf{Remark}\) We haven’t yet discussed how to solve second-order linear nonhomogeneous differential equations, but we don’t need to know how to do that in order to solve this problem.
Consider the linear differential equation
\[y'' - \frac{1}{z}y' + \frac{3}{z}y = 0 \ , z > 0.\] Let \(y_1\) and \(y_2\) be the solutions of the differential equation satisfying the initial conditions \(y_1(1) = 0, y'_1(1) = 0, y_2(1) = 0, y'_2(1) = 1.\) Let \(W\) be the Wronskian \(W(z) = y_1(z)y'_2(z) - y'_1(z)y_2(z).\)
What is \(W(1)\)? Without solving the differential equation explicitly, find an exact formula for \(W(z)\) (for general \(z>0\)).
Find the Wronskian of \(w_1, w_2\) where
(a) \(w_1 = \sin(t) \ , w_2 = \sin(t)\log(t)\); is the Wronskian defined for all \(t\)?
(b) \(w_1 = x^2 \ , w_2 = \log(x)\).
The Wronskian of two functions \(w_1(u)\) and \(w_2(u)\) equals \(\log(u)\). Can \(w_1\) and \(w_2\) be solutions of a differential equation \(w'' + p(u)w' + q(u)w = 0\), where \(p(u)\) and \(q(u)\) are continuous on the interval \((\frac{1}{2}, \frac{3}{2})\)? You should include complete sentences in your justification.
Find the Wronskian of the pair of functions \(\beta\cos^2(\theta), \alpha\left(1+\cos(2\theta)\right)\) and determine the value(s) of \(\alpha, \beta\) for which the Wronskian is nonzero. Can you think of a possible explanation for why you obtain that particular answer?
Show that if \(w = f(u)\) is a solution of the differential equation \(w'' + p(u)w' + q(u)w = g(u)\), where \(g(u)\) is not always \(0\), then \(w = cf(u)\), where \(c\) is any constant other than \(1\), is not a solution. Why this doesn’t contradict the principle of linear superposition?
For what values of the parameter \(\gamma\) are the characteristic roots to the following differential equation (a) distinct real, (b) real with multiplicity, or (c) complex conjugates?
\[y'' + \gamma y' + 1 = 0.\] In each case, write down a general solution to the corresponding differential equation.