Practice sheet

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Which function is a solution of $\dfrac{dy}{dx} = 5y$ ?
1. $y = 3e^{ 5x}$
2. $y = e^{3x}$
3. $y = 5xy$
4. $y = e^{x} + 5$
Which function is a solution of $\dfrac{dy}{dx} = 6y$ ?
1. $y = 3e^{ 6x}$
2. $y = e^{3x}$
3. $y = 6xy$
4. $y = e^{x} + 6$
Which function is a solution of $\dfrac{dy}{dx} = 9y$ ?
1. $y = 3e^{ 9x}$
2. $y = e^{3x}$
3. $y = 9xy$
4. $y = e^{x} + 9$
Which function is a solution of $\dfrac{dy}{dx} = 2y$ ?
1. $y = 3e^{ 2x}$
2. $y = e^{3x}$
3. $y = 2xy$
4. $y = e^{x} + 2$
Which function is a solution of $\dfrac{dy}{dx} = 7y$ ?
1. $y = 3e^{ 7x}$
2. $y = e^{3x}$
3. $y = 7xy$
4. $y = e^{x} + 7$
Which function is a solution of $\dfrac{dy}{dx} = 4y$ ?
1. $y = 3e^{ 4x}$
2. $y = e^{3x}$
3. $y = 4xy$
4. $y = e^{x} + 4$
Which function is a solution of $\dfrac{dy}{dx} = 8y$ ?
1. $y = 3e^{ 8x}$
2. $y = e^{3x}$
3. $y = 8xy$
4. $y = e^{x} + 8$
For which $k > 0$ is $y = e^{kx}$ a solution of $y'' - 9y = 0$ ?
$\dfrac{dy}{dt} = 5(y - 3)$ . Find the constant (equilibrium) solution $y = c$ .
In the slope field of $\dfrac{dy}{dx} = x + 3y$ , what is the slope of the segment at $(-3, 1)$ ?
In the slope field of $\dfrac{dy}{dx} = 9x^2$ , which segments are all parallel?
1. Those along any vertical line $x = c$
2. Those along any horizontal line $y = c$
3. None: every segment has its own slope
In the slope field of $\dfrac{dy}{dx} = 2x^2$ , which segments are all parallel?
1. Those along any vertical line $x = c$
2. Those along any horizontal line $y = c$
3. None: every segment has its own slope
In the slope field of $\dfrac{dy}{dx} = 4x^2$ , which segments are all parallel?
1. Those along any vertical line $x = c$
2. Those along any horizontal line $y = c$
3. None: every segment has its own slope
In the slope field of $\dfrac{dy}{dx} = 6x^2$ , which segments are all parallel?
1. Those along any vertical line $x = c$
2. Those along any horizontal line $y = c$
3. None: every segment has its own slope
In the slope field of $\dfrac{dy}{dx} = 5x^2$ , which segments are all parallel?
1. Those along any vertical line $x = c$
2. Those along any horizontal line $y = c$
3. None: every segment has its own slope
In the slope field of $\dfrac{dy}{dx} = 7x^2$ , which segments are all parallel?
1. Those along any vertical line $x = c$
2. Those along any horizontal line $y = c$
3. None: every segment has its own slope
In the slope field of $\dfrac{dy}{dx} = 8x^2$ , which segments are all parallel?
1. Those along any vertical line $x = c$
2. Those along any horizontal line $y = c$
3. None: every segment has its own slope
In the slope field of $\dfrac{dy}{dx} = 3x^2$ , which segments are all parallel?
1. Those along any vertical line $x = c$
2. Those along any horizontal line $y = c$
3. None: every segment has its own slope
Solve $\dfrac{dy}{dx} = \dfrac{x}{y}$ with $y(0) = 8$ (so $y > 0$ ). Give $y$ in terms of $x$ .
Solve $\dfrac{dy}{dx} = 4xy$ with $y(0) = 5$ .
Solve $\dfrac{dy}{dx} = y^2$ with $y(0) = \dfrac{1}{ 3 }$ .
A population grows with $\dfrac{dP}{dt} = kP$ . It starts at 104 and doubles every 9 years. What is it after 27 years?
Solve $\dfrac{dy}{dt} = -2y$ with $y(0) = 5$ .
A substance decays with $\dfrac{dm}{dt} = km$ and has a half-life of 27 years. Find $k$ (exact).
A drink at 65 °C cools in a 18 °C room: $\dfrac{dT}{dt} = -\tfrac{ 3 }{10}(T - 18)$ . Find $T(t)$ .
Soup at 60 °C is left in a 17 °C kitchen and follows Newton's law of cooling. Which temperature does it approach in the long run?
$\dfrac{dT}{dt} = -\tfrac{ 4 }{10}(T - 25)$ and $T(0) = 74$ . How fast is the temperature changing at $t = 0$ , in °C/min?

Answers

$y = 3e^{ 5x}$
$3$
$3$
$0$
Those along any vertical line $x = c$
$\sqrt{x^{2} + 64}$
$5 e^{2 x^{2}}$
$\frac{1}{3 - x}$
$832$
$5 e^{- 2 t}$
$- \frac{\ln{\left(2 \right)}}{27}$
$18 + 47 e^{- \frac{3 t}{10}}$
$17$
$- \frac{98}{5}$