Initial-Value Problem Integration Question

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In summary, Bob is working on a maths uni pre course and is seeking help with solving an equation involving differentiation and integration. The equation is dy/dx=(cos(3x))/(2-sin(3x)) and he is looking for pointers on how to approach it. The solution involves separating variables and making a u-substitution.
  • #1
bobbles22
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Hi there,

I'm doing a maths uni pre course with some questions on differentiation and integration.

I'm looking for a few pointers on how to proceed.

I need to solve:

dy/dx=(cos(3x))/(2-sin(3x)) when y=2 and x=0

Can anyone help with how I'm meant to approach this and how I go about actually integrating the equation. Its been some time since I've done things like this and the information I have is not very easy to understand.

Many thanks

Bob
 
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  • #2
you got:

[tex]\frac{dy}{dx}=\frac{\cos(3x)}{2-\sin(3x)}[/tex]

You can now separate variables and integrate:

[tex]\int_{y_0}^y dy=\int_{x_0}^x \frac{\cos(3x)}{2-\sin(3x)}dx[/tex]

The left side is just y-y_0. The other side you can do right? Just make a u-substitution. For example, what is the derivative of the denominator? Isn't that real close to what's in the numerator? Can you adjust it to be just right then integrate?
 

FAQ: Initial-Value Problem Integration Question

1. What is an Initial-Value Problem (IVP)?

An IVP is a type of mathematical problem that involves finding a solution to a differential equation, given an initial condition. This initial condition specifies the value of the dependent variable at a particular point in the independent variable's domain.

2. What is the purpose of solving an IVP?

Solving an IVP allows us to determine the behavior of a system over time. This can be useful in predicting future outcomes or understanding the dynamics of a process.

3. How do you solve an IVP?

To solve an IVP, we use a method called integration. This involves finding an antiderivative of the given differential equation and using the initial condition to determine the value of the constant of integration. The resulting solution is called the particular solution to the IVP.

4. What is the difference between a general solution and a particular solution?

A general solution is a solution to a differential equation that contains one or more arbitrary constants. These constants can take on any value, so the general solution represents all possible solutions to the equation. A particular solution, on the other hand, is a specific solution that is obtained by using initial conditions to determine the values of the arbitrary constants in the general solution.

5. What are some real-world applications of solving IVPs?

Solving IVPs is essential in many fields of science and engineering, such as physics, chemistry, biology, and economics. It can be used to model and predict the behavior of physical systems, chemical reactions, biological processes, and economic systems. For example, IVPs are used in weather forecasting, population growth analysis, and predicting the stock market trends.

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