Help with a simple initial value problem

In summary, to solve the initial value problem dy/dt=4y-2, y(0)=3, you can use separation of variables by rewriting the equation as dy/(4y-2)= dt and then integrating both sides with respect to y and t respectively. This will give you the solution y= 2y^2 - 2y + C, where C is a constant that can be determined by plugging in the initial conditions.
  • #1
bengaltiger14
138
0

Homework Statement



Solve the IVP dy/dt=4y-2, y(0)=3


This is how I work it. I integrate both sides and get y= 2y^2 - 2y + C

I then solve for C: 3=2(0)^2 - 2(0) + C

C = 3

Is this the correct way to solve for C?
 
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  • #2
[tex]\frac{dy}{dt}=4y-2[/tex]

[tex] \frac{1}{4y-2}dy= 1 dt[/tex]

Separation of variables. Now integrate.
 
  • #3
bengaltiger14 said:

Homework Statement



Solve the IVP dy/dt=4y-2, y(0)=3


This is how I work it. I integrate both sides and get y= 2y^2 - 2y + C

I then solve for C: 3=2(0)^2 - 2(0) + C

C = 3

Is this the correct way to solve for C?
It's not solving for C that is the problem! You have integrated on the left side with respect to t and on the right with respect to y. You can't do that. In order to integrate with respect to t you have to "multiply" by dt: (dy/dt) dt= dy, but then the right side is (4y- 2)dt which is NOT the same as (4y- 2)dy.

What you can do, as rock.freak667 suggested is "separate" the equation so you have only y on one side and t on the other:
dy= (4y-2)dt so dy/(4y-2)= dt. NOW you can integrate the left side with respect to y (because of the "dy") and the right side with respect to t (because of the "dt"). After you have done that you can set y= 3, t= 0 to find C, the constant of integration.
 

What is an initial value problem?

An initial value problem is a type of mathematical problem that involves finding a function that satisfies a given set of conditions called initial values. These initial values usually include the value of the function at a certain point and the value of its derivative at that point.

What is the purpose of solving an initial value problem?

The purpose of solving an initial value problem is to find the specific function that satisfies the given conditions. This is important in various fields of science, such as physics and engineering, where understanding and predicting the behavior of systems is crucial.

What are the steps for solving an initial value problem?

The steps for solving an initial value problem include:

  • 1. Identifying the given initial values.
  • 2. Finding the general solution of the differential equation.
  • 3. Plugging in the initial values to find the specific solution.

What are the common methods for solving an initial value problem?

There are several methods for solving an initial value problem, including Euler's method, Runge-Kutta method, and shooting method. The choice of method depends on the complexity of the problem and the desired level of accuracy.

What are some applications of initial value problems in science?

Initial value problems have various applications in science, including predicting the motion of objects in physics, modeling chemical reactions in chemistry, and analyzing population growth in biology. They are also used in engineering to design and optimize systems such as circuits and structures.

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