4.1 Basics of differential equations  (Page 6/14)

Find the particular solution to the differential equation $y^{\prime }=4x^2$ that passes through $(\mathrm{-3},\mathrm{-30}),$ given that $y=C+\frac{4x^3}{3}$ is a general solution.

Find the particular solution to the differential equation $y^{\prime }=3x^3$ that passes through $(1,4.75),$ given that $y=C+\frac{3x^4}{4}$ is a general solution.

Find the particular solution to the differential equation $y^{\prime }=3x^{2}y$ that passes through $(0,12),$ given that $y=C{e}^{x^3}$ is a general solution.

Find the particular solution to the differential equation $y^{\prime }=2xy$ that passes through $\left(0,\frac{1}{2}\right),$ given that $y=C{e}^{x^2}$ is a general solution.

Find the particular solution to the differential equation $y^{\prime }={\left(2xy\right)}^2$ that passes through $\left(1,-\frac{1}{2}\right),$ given that $y=-\frac{3}{C+4x^3}$ is a general solution.

Find the particular solution to the differential equation $y^{\prime }{x}^2=y$ that passes through $\left(1,\frac{2}{e}\right),$ given that $y=C{e}^{\text{−}1\text{/}x}$ is a general solution.

Find the particular solution to the differential equation $8\frac{dx}{dt}=\mathrm{-2}\text{cos}(2t)-\text{cos}(4t)$ that passes through $\left(\pi ,\pi \right),$ given that $x=C-\frac{1}{8}\text{sin}(2t)-\frac{1}{32}\text{sin}(4t)$ is a general solution.

Find the particular solution to the differential equation $\frac{du}{dt}=\text{tan}u$ that passes through $\left(1,\frac{\pi }{2}\right),$ given that $u={\text{sin}}^{\mathrm{-1}}\left(e^{C+t}\right)$ is a general solution.

Find the particular solution to the differential equation $\frac{dy}{dt}=e^{(t+y)}$ that passes through $\left(1,0\right),$ given that $y=\text{−}\text{ln}(C-e^t)$ is a general solution.

Find the particular solution to the differential equation $y^{\prime }(1-x^2)=1+y$ that passes through $\left(0,\mathrm{-2}\right),$ given that $y=C\frac{\sqrt{x+1}}{\sqrt{1-x}}-1$ is a general solution.

$y^{\prime }=3x+e^x$

$y^{\prime }=\text{ln}x+\text{tan}x$

$y^{\prime }=\text{sin}x{e}^{\text{cos}x}$

$y^{\prime }=4^x$

$y^{\prime }={\text{sin}}^{\mathrm{-1}}\left(2x\right)$

$y^{\prime }=2t\sqrt{t^2+16}$

$x^{\prime }=\text{coth}t+\text{ln}t+3t^2$

$x^{\prime }=t\sqrt{4+t}$

$y^{\prime }=y$

$y^{\prime }=\frac{y}{x}$

$\frac{dy}{dt}=2t$

$\frac{dy}{dt}=\text{−}t$

$\frac{dy}{dt}=2y$

$\frac{dy}{dt}=\text{−}y$

$\frac{dy}{dt}=2$

$\frac{dy}{dt}=4t$

$\frac{dy}{dt}=4y$

$\frac{dy}{dt}=\mathrm{-2}y$

$\frac{dy}{dt}=e^{4t}$

$\frac{dy}{dt}=e^{\mathrm{-4}t}$

[T] $y^{\prime }=y(x)$

[T] $x{y}^{\prime }=y$

[T] $y^{\prime }=t^3$

[T] $y^{\prime }=x+y$ ( Hint: $y=C{e}^x-x-1$ is the general solution)

[T] $y^{\prime }=x\text{ln}x+\text{sin}x$

Find the general solution to describe the velocity of a ball of mass $1\text{lb}$ that is thrown upward at a rate $a$ ft/sec.

In the preceding problem, if the initial velocity of the ball thrown into the air is $a=25$ ft/s, write the particular solution to the velocity of the ball. Solve to find the time when the ball hits the ground.

You throw two objects with differing masses $m_1$ and $m_2$ upward into the air with the same initial velocity $a$ ft/s. What is the difference in their velocity after $1$ second?

[T] You throw a ball of mass $1$ kilogram upward with a velocity of $a=25$ m/s on Mars, where the force of gravity is $g=\mathrm{-3.711}$ m/s ² . Use your calculator to approximate how much longer the ball is in the air on Mars.

[T] For the previous problem, use your calculator to approximate how much higher the ball went on Mars.

[T] A car on the freeway accelerates according to $a=15\text{cos}(\pi t),$ where $t$ is measured in hours. Set up and solve the differential equation to determine the velocity of the car if it has an initial speed of $51$ mph. After $40$ minutes of driving, what is the driver’s velocity?

[T] For the car in the preceding problem, find the expression for the distance the car has traveled in time $t,$ assuming an initial distance of $0.$ How long does it take the car to travel $100$ miles? Round your answer to hours and minutes.

[T] For the previous problem, find the total distance traveled in the first hour.

Substitute $y=B{e}^{3t}$ into $y^{\prime }-y=8e^{3t}$ to find a particular solution.

Substitute $y=a\text{cos}(2t)+b\text{sin}(2t)$ into $y^{\prime }+y=4\text{sin}(2t)$ to find a particular solution.

Substitute $y=a+bt+c{t}^2$ into $y^{\prime }+y=1+t^2$ to find a particular solution.

Substitute $y=a{e}^t\text{cos}t+b{e}^t\text{sin}t$ into $y^{\prime }=2e^t\text{cos}t$ to find a particular solution.

Solve $y^{\prime }=e^{kt}$ with the initial condition $y(0)=0$ and solve $y^{\prime }=1$ with the same initial condition. As $k$ approaches $0,$ what do you notice?