7. Antiderivatives & Indefinite Integrals

Solve the initial value problem using the method of Laplace transforms: $y^{\mathrm{\text{'}\text{'}}}+4y=0$, $y\left(0\right)=2$, $y^{\mathrm{\text{'}}}\left(0\right)=0$. What is $y\left(t\right)$?

Solve the following initial-value problem using Laplace transforms: $y^{\text{'}\text{'}}+4y=0$, $y\left(0\right)=2$, $y^{\text{'}}\left(0\right)=0$. What is $y\left(t\right)$?

Solve the initial-value problem: $x(x+1)\frac{dy}{dx}+xy=1$, $y\left(e\right)=1$. What is the solution $y\left(x\right)$?

Solve the initial value problem: The differential equation is homogeneous. $x{y}^2\frac{dy}{dx}=y^3-x^3$, $y\left(1\right)=4$. What is the explicit solution $y\left(x\right)$?

Solve the initial value problem for the homogeneous differential equation $(x^2+2y^2)\frac{dx}{dy}=xy$, with the initial condition $y(-1)=2$. What is the explicit solution $y\left(x\right)$?

Solve the initial value problem: $x^{\prime }=2x$, $x\left(0\right)=4$. What is $x\left(t\right)$?

Consider the initial value problem: $y^{\mathrm{\text{'}\text{'}}}+16y=\cos \left(4t\right)$, with $y\left(0\right)=0$ and $y^{\mathrm{\text{'}}}\left(0\right)=0$. What is the particular solution $y\left(t\right)$?

Solve the following initial value problem:

$\frac{dy}{dx}=2x-5$; $y\left(0\right)=4$

Using the acceleration function below, find the velocity function, if the velocity is v = 5 at time t = 2.

$a\left(t\right)=-20$

Find the function $f\left(x\right)$ that satisfies the following differential equation.

$f^{\prime\prime}\left(x\right)=3x^2$; $f^{\prime}\left(0\right)=1$; $f\left(1\right)=3$

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .

a. Find the velocity of the object for all relevant times. 

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .

d. Find the time when the object strikes the ground.

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .

b. Find the position of the object for all relevant times. 

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .

c. Find the time when the object reaches its highest point. What is the height? 

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.

v(t) = 2t + 4; s(0) = 0