L3 Differentiation
Question Bank
Show that $y = 2e^{-5x} + 3$ satisfies the differential equation
$ \frac{dy}{dx} + 5y = 15 $
Show that $y = e^{-x}\sin 2x$ satisfies the differential equation
$ y'' + 2y' + 5y = 0 $
Show that $y = x^2e^{-1/x},$ $x \ne 0,$ satisfies the differential equation
$ x^2\frac{dy}{dx} = (2x + 1)y $
Show that $y = x^2e^{-x}$ satisfies the differential equation
$ y'' + 2y' + y = 2e^{-x} $
Show that $y = \dfrac{\sin x}{x},$ $x > 0,$ satisfies the differential equation
$ xy'' + 2y' + xy = 0 $
Find all the values of the constant $k$ for which $y = x^k,$ $x > 0,$ satisfies the differential equation
$ x^2y'' - 2xy' - 4y = 0 $
The curvature, $\kappa,$ of a curve $y = f(x)$ is given by
$ \kappa = \frac{\dfrac{d^2y}{dx^2}}{\left(1 + \left(\dfrac{dy}{dx}\right)^2\right)^{\frac{3}{2}}} $
Prove that the curvature of the curve $y = \ln(\sec x),$ $-\dfrac{\pi}{2} < x < \dfrac{\pi}{2},$ is $\kappa = \cos x$ at every point.
The function $y = e^{ax}\sin bx,$ where $a$ and $b$ are constants and $b > 0,$ satisfies the differential equation
$ y'' - 4y' + 13y = 0 $
for all values of $x.$ Find the values of $a$ and $b.$
Show that
$ y = A\cos(\ln x) + B\sin(\ln x) $
where $x > 0$ and $A$ and $B$ are constants, satisfies the differential equation
$ x^2y'' + xy' + y = 0 $
for all values of $A$ and $B.$