Put the differential equation \[\frac{t}{e^t + 1}\, y'''(t) + 3ty'(t) = \cos(t)\] into normal form. What is its order?
For #2 – #7, state the largest interval on which a solution to the given initial value problem is determined by the conditions given.
\(y'' + 5y' + 3y = e^t\), \(y(0) = 0\), \(y'(0) = 1\).
\(w\ddot{z} + 5\dot{z} + 3z = e^w\) , \(z(1) = 0\) , \(\dot{z}(1) = 1\) .
\(y''' + \frac{1}{t}\,y'' + \tan(t)y' + y = \frac{t}{t^2 - 4}\), \(y(3) = 0\), \(y'(3) = -1\), \(y''(3) = 2\pi\).
\((x^2 - 1)\ddot{y} + \frac{y}{x-3} = e^x\cos(x)\) , \(y(2) = \dot{y}(2) = \pi\) .
\(y'' + a(t)y' + b(t) = 0\), \(y(3) = y'(3) = 0\), where
\[a(t) = \left\{ \begin{array}{rc} 1 & t \leq 2 \\ -1 & t > 2 \end{array} \right.\] and \[b(t) = \left\{ \begin{array}{rc} 0 & t < 0 \\ 1 & 0 \leq t \leq 5 \\ 2 & t > 5\text{.} \end{array} \right.\]
\(z'' + c(w)z = \sec(w)\) , \(z(1) = 1\) , \(z'(1) = -3\) , where \[c(w) = \left\{\begin{array}{rc} w & , w < 0 \\ w^2 & , w >0\text{.} \end{array} \right.\]
Can \(x^3\) be the solution to a differential equation of the form \(w'' + aw' + bw = 0\), where \(a\) and \(b\) are real constants? Explain why or why not.
What is special about the exponent \(3\) in part (a)? Suggest a generalization, and explain why it’s correct.
Show that \(f(t) = e^t + e^{2t} + e^{3t}\) cannot be the solution of any differential equation of the form \(\Dop^2y + a\Dop y + by = c\), where \(a\), \(b\), and \(c\) are constants. [Hint. Show that if you plug in \(f\) into the equation, then the left side (\(f'' + af' + bf\)) must be unbounded for large \(t\), regardless of the choice of \(a\) or \(b\).]
Check that, for any real constants \(a\) and \(b\), the function \(ae^t + be^{-t}\) is a solution to the differential equation \(\Dop^2y - y = 0\).
Try to find values of \(a\) and \(b\) so that the function \(ae^t + be^{-t}\) satisfies the initial conditions \(y(0) = y'(0) = 0\), or prove it can’t be done.
Try to find values of \(a\) and \(b\) so that the function \(ae^t + be^{-t}\) satisfies the initial conditions \(y(0) = 1\), \(y'(0) = -1\), or prove it can’t be done. [Hint. Later we will see that in fact there’s a way to satisfy any initial conditions by appropriately choosing values of \(a\) and \(b\). So it can be done.]
Put the differential equation
\[(u^2 + 4)x'''' + 2ux' + 2x = \sin(u)\] into normal form. What is its order? Is it homogeneous or not?
Put the differential equation \[y''' + \frac{3}{x}y' + \frac{\cos(5x)}{4 +x }y = \frac{e^x}{x - 3}\] into normal form. What is its order? Is it homogeneous or not?
Find the largest interval \(a < x < b\) on which the initial-value problem
\[\cos(x)y'' + \sin(x)y = x,\ y(1) = -1,\ y'(1) = 1.\] can be guaranteed to have a solution.
State the largest interval on which a solution to the given initial value problem is determined by the conditions given:
\((u^2 - 9)x'' + 3x = \log (|30 - 4u|) \) , \(x(5) = -3\) , \(x'(5) = 2\) .