Solving the Initial Value Problem for 'y' = x, x' = -5y-4x

In summary, solving the initial value problem for the system of differential equations 'y' = x and x' = -5y-4x can be done by first finding the general solution, then applying the initial conditions to find the particular solution. This involves solving for the constants of integration and plugging them into the general solution, resulting in a unique solution that satisfies the given initial conditions. The method used for solving this problem is known as the method of undetermined coefficients, which is a common technique in solving linear first-order differential equations.
  • #1
Ry122
565
2
y' = x
x' = -5y-4x

y(0) = 1
x(0) = 0

after finding the general solution as shown here
http://www.wolframalpha.com/input/?i=y'+=+x,+x'+=+-5y-4x

how do you go about applying the initial values and finding the complete solution?
 
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  • #2
Your general solution should have two undetermined coefficients. Substitute 0 for t, set x= 0, y= 1 and you will have two equations to solve for the two coefficients.
 
  • #3
actually I don't think wolfram alpha has done the correct thing in making x and y a function of t as there's no mention of another variable in the original equations. What would you do when they aren't functions of t?
 
  • #4
You can call the independent variable whatever you want! What did you mean by x' and y'? I assumed the primes were derivatives. With respect to what variable?
 

FAQ: Solving the Initial Value Problem for 'y' = x, x' = -5y-4x

What is the Initial Value Problem for 'y' = x, x' = -5y-4x?

The Initial Value Problem for 'y' = x, x' = -5y-4x refers to the mathematical equation that describes the relationship between the variables y and x. It is used to find the values of y and x at a specific starting point, or initial value, given the differential equation y' = x and the initial condition x' = -5y-4x.

How is the Initial Value Problem solved for 'y' = x, x' = -5y-4x?

The Initial Value Problem for 'y' = x, x' = -5y-4x can be solved using various methods, such as separation of variables, variation of parameters, and Laplace transforms. These methods involve manipulating the differential equation to isolate the variables and solving for the initial conditions to find the values of y and x.

What are the applications of solving the Initial Value Problem for 'y' = x, x' = -5y-4x?

Solving the Initial Value Problem for 'y' = x, x' = -5y-4x has many practical applications in fields such as physics, engineering, and economics. It can be used to model and predict the behavior of systems that involve changing variables, such as population growth, radioactive decay, and electrical circuits.

How does the solution of the Initial Value Problem change with different initial conditions?

The solution of the Initial Value Problem for 'y' = x, x' = -5y-4x can vary depending on the initial conditions given. The initial conditions determine the starting point for the solution and can affect the behavior of the system being modeled. Changing the initial conditions can result in different values for y and x, and can also alter the shape of the solution curve.

What are the limitations of solving the Initial Value Problem for 'y' = x, x' = -5y-4x?

While solving the Initial Value Problem for 'y' = x, x' = -5y-4x can provide valuable insights and predictions, it also has its limitations. The solution may not accurately reflect the behavior of the system in the long term, as it is based on a single starting point. Additionally, the accuracy of the solution can be affected by the assumptions and simplifications made in the mathematical model.

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