Initial Value Problem for (DE)

In summary, the problem involves finding the time it takes for an object to reach 98% of its limiting velocity and the distance it falls in that time. To solve this, the given differential equation must be rewritten in a specific form before integrating both sides. The final solution involves the natural logarithm function.
  • #1
Vanessa Avila
94
1

Homework Statement


dv/dt = 9.8 - (v/5) , v(0) = 0

(a) The time it must elapse for the objet to reach 98% of its limiting velocity
(b) How far does the object fall in the time found in part (a)?

Homework Equations


(dv/dt)/(9.8-(v/5))

The Attempt at a Solution


I'm a little overwhelmed by this class and I think the problem I have is I'm not catching on as to why the next answer is the way it is, so it would be nice if someone could explain to me why.
As I read on my textbook, I'm supposed to rewrite the form of the eqn first in which I attempted to do:

(dv/dt)/(9.8-(v/5)) = 1dt
but why do we have to rewrite the eqn in this form? I checked the solution for this, and it said it was right. I know afterwards you then integrate both sides, but I saw that they got
−5ln(9.8−(v/5))=t+C
I understand where the t+C comes from but not where the -5ln(9.8-(v/5)) comes from. Can someone explain to me how to get to that?
 
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  • #2
What is the integral of 1/x?
 
  • #3
Orodruin said:
What is the integral of 1/x?
ln(x)! Ah I missed that. Thanks!
 

Related to Initial Value Problem for (DE)

What is an initial value problem (IVP) for a differential equation (DE)?

An initial value problem for a differential equation is a mathematical problem that involves finding a solution to a differential equation that satisfies specified initial conditions. These conditions usually include the value of the function at a specific point and the rate of change of the function at that point.

Why is solving an initial value problem for a differential equation important?

Solving an initial value problem for a differential equation is important because it allows us to predict the behavior of a system over time. This is essential in many scientific fields, such as physics, engineering, and economics, where the behavior of a system is described by a differential equation.

What are the steps involved in solving an initial value problem for a differential equation?

The steps involved in solving an initial value problem for a differential equation are:

  1. First, the differential equation is rewritten in its standard form.
  2. Next, the initial conditions are substituted into the differential equation to obtain a system of equations.
  3. The system of equations is then solved to find the values of the constants in the solution.
  4. Finally, the solution is obtained by substituting the values of the constants back into the general solution of the differential equation.

What are some common techniques used to solve initial value problems for differential equations?

Some common techniques used to solve initial value problems for differential equations include separation of variables, variation of parameters, and Laplace transforms. These techniques allow us to find exact solutions to differential equations, rather than just approximate solutions.

Can initial value problems for differential equations have multiple solutions?

Yes, initial value problems for differential equations can have multiple solutions. In fact, some differential equations have an infinite number of solutions. However, when solving an initial value problem, we are usually looking for the unique solution that satisfies the given initial conditions.

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