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AHL 5.14—Setting up a DE, solve by separating variables

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Topic 5—Calculus » AHL 5.14—Setting up a DE, solve by separating variables

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Directly related questions

21M.2.AHL.TZ1.7a:
Find the population of rabbits $1$ $1$ year after they were introduced.
21M.3.AHL.TZ2.2d.iii:
Alessia estimates that the mackerel population density increases by a factor of three every two years. Show that $α = 0.549$ $α = 0.549$ to three significant figures.
21M.3.AHL.TZ2.2d.ii:
Show that the expression for the mackerel population density after $t$ $t$ years is $M = M_{0} e^{α t}$ $M = M_{0} e^{α t}$
21M.3.AHL.TZ2.2d.i:
Write down the differential equation for $M$ $M$ that models this situation.
21M.1.AHL.TZ1.12a:
Show that $V = {(20 - \frac{t}{5})}^{2}$ $V = {(20 - \frac{t}{5})}^{2}$ .
21M.1.AHL.TZ1.12b:
Find the time taken for the tank to empty.
21N.3.AHL.TZ0.2a.i:
Find the equation of the regression line of $h$ $h$ on $t$ $t$ .
21N.3.AHL.TZ0.2a.iii:
Suggest why Eva’s use of the linear regression equation in this way could be unreliable.
21N.3.AHL.TZ0.2b.i:
Find the equation of the least squares quadratic regression curve.
21N.3.AHL.TZ0.2b.ii:
Find the value of $k$ $k$ .
21N.3.AHL.TZ0.2b.iii:
Hence, write down a suitable domain for Eva’s function $h (t) = p t^{2} + q t + r$ $h (t) = p t^{2} + q t + r$ .
21N.3.AHL.TZ0.2a.ii:
Interpret the meaning of parameter $a$ $a$ in the context of the model.
21N.3.AHL.TZ0.2d:
By solving the differential equation $\frac{d h}{d t} = - R^{2} \sqrt{70560 h}$ $\frac{d h}{d t} = - R^{2} \sqrt{70 560 h}$ , show that the general solution is given by $h = 17640 {(c - R^{2} t)}^{2}$ $h = 17 640 {(c - R^{2} t)}^{2}$ , where $c \in ℝ$ .
21N.3.AHL.TZ0.2g.i:
Show that $\frac{d H}{d t} \approx 0.2514 - 0.009873 t - 0.1405 \sqrt{H}$ , where $0 \leq t \leq T$ .
21N.3.AHL.TZ0.2e:
Use the general solution from part (d) and the initial condition $h (0) = 3.2$ to predict the value of $T$ .
21N.3.AHL.TZ0.2f:
Find this new height.
21N.3.AHL.TZ0.2g.ii:
Use Euler’s method with a step length of $0.5$ minutes to estimate the maximum value of $H$ .
21N.3.AHL.TZ0.2c:
Show that $\frac{d h}{d t} = - R^{2} \sqrt{70 560 h}$ .
SPM.2.AHL.TZ0.7b:
From your solution to part (a), or otherwise, find the terminal velocity of the object predicted by this model.
SPM.2.AHL.TZ0.7f:
Use the differential equation to find the terminal velocity for the object.
SPM.2.AHL.TZ0.7d:
Use Euler’s method, with a step length of 0.2, to find the displacement and velocity of the object when $t = 0.6$ .
SPM.2.AHL.TZ0.7e:
By repeated application of Euler’s method, find an approximation for the terminal velocity, to five significant figures.
SPM.2.AHL.TZ0.7g:
Use your answers to parts (d), (e) and (f) to comment on the accuracy of the Euler approximation to this model.
SPM.2.AHL.TZ0.7c:
Write down the differential equation as a system of first order differential equations.
22M.3.AHL.TZ1.1b.i:
Find the general solution of the differential equation $Q' (t) = β N Q (t)$ .
SPM.2.AHL.TZ0.7a:
By substituting $v = \frac{{{\text{d}}x}}{{{\text{d}}t}}$ into the equation, find an expression for the velocity of the particle at time $t$ . Give your answer in the form $v = f(t)$ .
16N.2.AHL.TZ0.H_6:
An earth satellite moves in a path that can be described by the curve $72.5{x^2} + 71.5{y^2} = 1$ where $x = x(t)$ and $y = y(t)$ are in thousands of kilometres and $t$ is time in seconds.

Given that $\frac{{{\text{d}}x}}{{{\text{d}}t}} = 7.75 \times {10^{ - 5}}$ when $x = 3.2 \times {10^{ - 3}}$ , find the possible values of $\frac{{{\text{d}}y}}{{{\text{d}}t}}$ .

Give your answers in standard form.

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