Particular Solution

Particular solution

Antiderivative

Start

Particular Solution

We know that the equation has an infinite number of solutions that differs only on the constant value C.

C = 3

C = 2

C = 1

C = 0

For example lets take de function so the general solution of the antiderivative will be:

C = -1

C = -2

C = -3

Particular Solution

So if we want the particular solution of the function that goes through the point (2,5), we use the general solution:

C = 3

C = 2

C = 1

C = 0

And we substitute the initial condition F(2) = 5 and we get the value of the constant.

C = -1

C = -2

C = -3

Example 1

Find the particular solution that satisfies the equation and the initial condition:

Example 1

Find the particular solution that satisfies the equation and the initial condition:

2. Then we susbstitute the initial condition

1. First we find the general solution:

3. The particular solution is when C = 8

Example 2

Find the particular solution that satisfies the equation and the initial condition:

Example 2

Find the particular solution that satisfies the equation and the initial condition:

3. At the end the first derivative is

so now we integrate the first derivative

1. First we integrate the second derivative:

4. We substitute the initial condition

2. We susbtitute the initial condition:

5. So the particular solution is:

Excercises

Find the particular solution that satisfies the next equations:

Applications of particular antiderivatives

Start

Applications

Given a function that describes the position of a moving object with respect of time

We know that the derivative over time of the position is the velocity

And the derivative with over time of the velocity is the acceleration

Applications

So now we can say that the integral of the acceleration over time is change in velocity

and the integral of the velocity over time is change in position

Example

A particle moves with an acceleration of with an initial velocity of -10 and initial position 0. Find the position of the function at t = 1.

Example

A particle moves with an acceleration of with an initial velocity of -10 and initial position 0. Find the position of the function at t = 1.

3. We find the integral of the velocity

to get the position function

1. First we integrate the acceleration to get the velocity function:

4. We susbtitute the initial conditions

2. We susbtitute the initial conditions

5. We susbtitute t = 1 in the position function

Exercise

The equation of acceleration of an object is given by a) Find the equation of velocity if after 1 second the velocity is 10 meters / second b) Find the equation of position if after 2 seconds the position is 12 meters

Example

A ball is thrown upward with a initial velocity of 60 ft/s from an initial position of 64 ft. Consider the gravity as 32 ft/seg2 a) What is the position function with respect of time t ? b) When does the ball will hit the ground? c) What is its velocity when it hits the ground? d) What is the maximun height it gets? The initial conditions of the problem are:

Example

A ball is thrown upward with a initial velocity of 60 ft/s from an initial position of 64 ft. Consider the gravity as 32 ft/seg2 a) What is the function of the position with respect of time t ?

Example

A ball is thrown upward with a initial velocity of 60 ft/s from an initial position of 64 ft. Consider the gravity as 32 ft/seg2 b) When does the ball will hit the ground?

Example

A ball is thrown upward with a initial velocity of 60 ft/s from an initial position of 64 ft. Consider the gravity as 32 ft/seg2 c) What is its velocity when it hits the ground?

Example

A ball is thrown upward with a initial velocity of 60 ft/s from an initial position of 64 ft. Consider the gravity as 32 ft/seg2 d) What is the maximun height it gets?

Exercises

1. An object is thrown vertically upward from an initial height of 5 fts and with a initial velocity of 48 ft/sec, whats is the maximum height it will reach? 2. We let fall an objet from a building with a height of 80 meters, how much time does it take to hit the ground and with what speed?