7. Antiderivatives & Indefinite Integrals

Initial Value Problems

Solve the initial value problems in Exercises 89–92.

d^3 r/dt^3 = - cos t; r''(0) = r'(0) = 0 , r(0) = -1

104–107. Functions from derivatives Find the function f with the following properties.

h'(x) = (x⁴ -2) /(1 + x²) ; h (1) = -(2/3) 

Solve the initial value problems in Exercises 55–58.

57. d²y/dx² = 2e^(−x),y(0) = 1,y′(0) = 0

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dy/dx = 2x − 7, y(2) = 0

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

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.

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 $y\left(x\right)$?

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.

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dv/dt = (1/2)sec t tan t, v(0) = 1

In Exercises 129–132 solve the initial value problem.

131. x dy - (y + √y)dx = 0, y(1) = 1

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

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

Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.

f(x) = 8x³ + sin x; F(0) = 2