Find the general solution to the following differential equations.
Consider the second-order differential equation \(x'' = x\). The four functions \(\cosh(u)\), \(\sinh(u)\), \(e^u\), and \(e^{-u}\) are all solutions to it. Is that weird? Why (or why not)?
Consider the fourth-order differential equation \(y^{(4)} = y\). Check that the functions \(y_1(t) = \cos(t)\) and \(y_2(t) = \sin(t)\) both are solutions to it. There should be two other fundamentally different solutions; can you find them by inspection?
Check that \(e^{3z}\) and \(e^{4z}\) are solutions to \(w'' - 7w' + 12w = 0\). Then find \(c_1\) and \(c_2\) so that \(W(z) = c_1e^{3z} + c_2e^{4z}\) is a solution to the differential equation satisfying the initial conditions \(W(0) = 2\) , \(W'(0) = 0\) .
Check that \(e^{x}\sin(x)\) and \(e^{x}\cos(x)\) are solutions to \(y'' - 2y' + 2y = 0\). Then find \(c_1\) and \(c_2\) so that \(Y(x) = c_1e^{x}\sin(x) + c_2e^{x}\cos(x)\) is a solution to the differential equation satisfying the initial conditions \(Y(0) = 3\), \(Y'(0) = -2\).
Check that \(e^{x}\) and \(x^2 + 2x + 2\) are solutions to the equation \[xy'' - (x+2)y' + 2y = 0\text{.}\]
Find \(c_1\) and \(c_2\) so that \(Y(x) = c_1e^{x} + c_2(x^2 + 2x + 2)\) is a solution to the differential equation satisfying the initial conditions \(Y(1) = e\), \(Y'(1) = e+2\).
On what interval(s) is the function \(Y\) you found in part (b) a solution to the equation?
Check that \(w\), \(w^3\), and \(w^5\) are solutions to the third-order differential equation
\[w^3z''' - 6w^2z'' + 15wz' - 15z = 0\text{ .}\] Then find constants \(c_1\), \(c_2\), and \(c_3\) so that \(Z(w) = c_1w + c_2w^3 + c_3w^5\) are solutions to the equation on \((0, \infty)\) satisfying \(Z(2) = 0\) , \(Z'(2) = 0\) , and \(Z''(2) = 1\).
One fundamental set of solutions for the differential equation \(y'' - 2y' - 8y = 0\) is \(\left\{e^{-2w}, e^{4w} \right\}\). Find a solution \(Y = c_1e^{-2w} + c_2e^{4w}\) which satisfies the general initial conditions \(Y(0) = y_0\) , \(Y'(0) = y_1\).
One fundamental set of solutions for the differential equation \(y'' + 9y = 0\) is \(\left\{ \cos(3t), \sin(3t) \right\}\). Find a solution \(Y = c_1\cos(3t) + c_2\sin(3t)\) that satisfies the general initial conditions \(Y(\frac{\pi}{2}) = y_0\), \(Y'(\frac{\pi}{2}) = y_1\).
One fundamental set of solutions for the differential equation \(u^2x'' + 4ux' = 0\) is \(\left\{1, u^{-3} \right\}\). Find a solution \(X(u) = c_1 + c_2u^{-3}\) that satisfies the general initial conditions \(X(1) = x_0\) , \(X'(1) = x_1\).
One fundamental set of solutions for the differential equation \(y'' - \frac{1}{x}y' = 0\) is \(\left\{1, x^2 \right\}\). Find a solution \(Y(x) = c_1 + c_2 x^2\) that satisfies the general initial conditions \(Y(-2) = y_0\), \(Y'(-2) = y_1\).
What is the interval of definition for your solution in (a)?
What happens if you try to solve for the initial conditions \(y(0) = 1\), \(y'(0) = 2\)?
Compute the Wronskian of the functions \(W_1(z) = z\) and \(W_2(z) = \cos(z)\). Is this defined for all points \(z\)?
Compute the Wronskian of the functions \(X_1(u) = \cosh(u)\) , \(X_2(u) = \sinh(u)\) , and \(X_3(u) = e^u\). Notice this also explains what was going on in Exercise #1.
(Continuation of Exercise #3) Show that \(\left\{e^{3t}, e^{4t}\right\}\) is a fundamental set of solutions for the second-order differential equation \(y'' -7y' +12y = 0\).
(Continuation of Exercise #4) Show that \(\left\{e^{x}\sin(x), e^{x}\cos(x)\right\}\) is a fundamental set of solutions for the second-order differential equation \(y'' -2y' +2y = 0\).
(Continuation of Exercise #6) Show that \(\left\{x, x^3, x^5 \right\}\) is
a fundamental set of solutions for the third-order differential equation \[x^3y''' - 6x^2y'' + 15xy' - 15y = 0\text{.}\]
Show that the functions \(e^t\), \(e^{2t}\), and \(e^{3t}\) are linearly independent on \((-\infty, \infty)\).
Show that \(\log(z)\) , \(\log(5z)\) , and \(1\) are linearly dependent over
\((0,\infty)\).
(Continuation of Exercise #3) Find a natural fundamental set for the differential equation \(y'' - 7y' + 12y = 0\) associated with the initial time \(0\).
(Continuation of Exercise #9) Find a natural fundamental set for the differential equation \(w^2z'' + 4wz' = 0\) associated with the initial time \(1\).
(Continuation of Exercise #8) Find a natural fundamental set for the differential equation \(y'' + 9y = 0\) associated with the initial time \(\frac{\pi}{2}\).
Check that \(e^z\) is a solution to \(w'' - 2w' + w = 0\), and then use reduction of order to find a fundamental set.
It’s clear that \(Y_1(t) = 1\) solves the differential equation \(ty'' + y' = 0\). Find another solution by reduction of order.
If \(n\) is a positive integer, then \(Y_1(t) = 1\) is also a solution to \(y'' - \frac{n}{x}\,y'= 0\). Complete a fundamental set for this equation. [Hint. Your answer will have an \(n\) in it somewhere.]
Check that \(Z_1(u) = e^{2u}\cos(u)\) is a solution to \(D^2z - 4Dz + 5z~=~0\). Then use reduction of order to complete a fundamental set for it.
Check that \(Z_1(w) = w\) is a solution to \((w-1)(w-2)\ddot{z} - w\ddot{z} + z~=~0\). Then use reduction of order to find a second linearly independent solution.
As suggested in the text, give a proof of Abel’s theorem for the third-order case. That is to say, suppose \(Y_1(t)\), \(Y_2(t)\), and \(Y_3(t)\) are three solutions to the differential equation
\[y'''(t) + a_1(t) y''(t) + a_2(t) y'(t) + a_3(t)y(t) = 0\text{,}\] then show that their Wronskian \(W(t) = W[Y_1, Y_2, Y_3](t)\) satisfies the first-order differential equation \[W' + a_1(t)W = 0\text{.}\]
Suppose we have a second-order homogeneous differential equation and we happen to know one of the solutions. Then the method of reduction of order will always give us a first-order differential equation whose solution is a linearly independent solution to the equation. In the problems above, the first-order differential equation is solvable, but this doesn’t happen in general—often we wind up with an integral that we can’t solve. This is not to say all is lost; having an integral (or even just a differential equation in the first place) opens up the possibility of determining values of the function by use of numerical methods, after all.
Consider the differential equation \[y'' - 2(2x^2 + 1)y = 0 \text{.}\] One of the homogeneous solutions is \(Y_1(x) = e^{x^2}\). Check this. Then use the method of reduction of order to come up with an expression for another solution to the homogeneous equation, linearly independent from \(Y_1\). [You should guess from the paragraph preceeding this that you probably will get stuck at an integral.]
Show that \(\left\{x^{-1}, x^{\frac{3}{2}} \right\}\) is a fundamental set of solutions for the second-order differential equation \(2x^2y'' + xy' - 3y = 0\). Make sure to check to see if the Wronskian of \(x_1\) and \(x_2\) is defined everywhere.
(Note: you will need first, to verify that each of the two functions \(\left\{x^{-1}, x^{\frac{3}{2}} \right\}\) satisfies the differential equation and second, to show that they form a fundamental set of solutions.)
Without solving, determine the Wronskian of two solutions evaluated at \(t=1\) for the following differential equation:
\[t^5\ddot{y} - 2t^2\dot{y} - t^7y = 0\ , y(1) = 5\ , \dot{y} =10.\] What is the interval of definition of this particular solution to the initial-value problem?
Without solving, determine the Wronskian of two solutions evaluated at \(u = 4\) for the following differential equation:
\[2u^2y'' + uy' - 3y = 0.\] Is the Wronskian defined for all \(u\)?
The following problem is an application of the Method of Linear Superposition, \(\bf{Theorem~2.1}\). Assume that \(\cos(x)\) and \(x\) are both solutions of the equation \(p(\Dop)w = q(x)\), for a certain polynomial \(p(x)\) and a certain function \(q(x)\).
(a) Write down a nonzero solution of the equation \(p(\Dop)w = 0\).
(b) Write down a solution \(w(x)\) of \(p(\Dop)w = q(x)\) such that \(w(0) = 2\).
\(\bf{Remark}\) We haven’t discussed how to solve nonhomogeneous second-order linear differential equations yet, but we don’t need to know that yet!
Prove the following statement: “If \(f(t)\) and \(g(t)\) are linearly independent solutions of a second-order linear homogeneous differential equation on an interval \(I\), then \(f\) and \(g\) cannot have a maximum at the same location in \(I\)."
(\(\bf{Hint:}\) This is a more theoretical argument than what you’ve seen before. Think of an argument by contradiction. Your proof should include complete sentences.)
Verify that \(y_1(x) = 1\) and \(y_2(x) = x^{\frac{1}{2}}\) are solutions to the differential equation \(yy'' + (y')^2 = 0\) for \(x > 0\). Then show that \(c_1 + c_2 x^{\frac{1}{2}}\) is not in general a solution of this equation. Can you explain why this result doesn’t contradict the Method of Linear Superposition in \(\bf{Theorem~2.1}\)?
\(\bf{More~exploration~of~the~Wronskian}\)
If the functions \(w_1\) and \(w_2\) are linearly independent solutions of \(w'' + p(u)w' + q(u)w = 0\), determine what the necessary and sufficient conditions are such that the functions \(w_3 = \alpha w_1 + \beta w_2\) and \(w_4 = \gamma w_1 + \epsilon w_2\) also form a linearly independent set of solutions.
\(\bf{A~clever~use~of~Abel's~Theorem:~}\)
(a) Consider the differential equation \(z'' + 2az' + a^2z = 0\) . Show that one of the solutions of the equation is \(e^{-au}\).
(b)Use Abel’s Formula to show that the Wronskian of any two solutions of the given equaton is \[W(u) = z_1(u)z'_2(u) - z'_1(u)z_2(u) = ce^{-2au} ,\] where \(c\) is a constant.
(c) Consider \(z_1(u) = e^{-au}\) from part (a) and use the result in (b) to obtain a differential equation satisfied by the second solution \(z_2(u)\). Then solve this equation to show that \(z_2(u) = ue^{-au} .\)
Use the method of order reduction to find a second solution of the differential equation \(z^2\ddot{w} + 2z\dot{w} -2w = 0 \ , z> 0 \ , w_1(z) = z\).