How Do You Solve This First Order ODE with Given Condition?

In summary, the given equation is rearranged and integrated to get the solution of y=-x/(Cx+1). Using the given condition, y(1)=3, the value of C is solved for and then substituted back into the equation to get the final solution of y=3x/(4x-3).
  • #1
mkay123321
16
0
The question is x^2dy/dx + y^2=0 , y(1)=3

I re-arrange the equation to get -1/y^2dy=1/x^2dx

Seperated them, then I integrate both sides to get 1/y=-1/x + c

Now I don't get how they got the answer y=3x/(4x-3), as when I try use the condition I get a different answer, could anyone help? I might have done something wrong in the integration?
 
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  • #2
You just solve it for y: [tex]y=\frac{-x}{Cx+1}[/tex]. Then set y(1) equal to 3 (since that's the condition you're given) and solve (using good ol' algebra) for C: [tex]y(1)=\frac{-1}{C+1}=3[/tex]
So, C=-4/3 and after you plug C into y, it is what your book says it is.
 

FAQ: How Do You Solve This First Order ODE with Given Condition?

1. What is a first order ODE with condition?

A first order ODE (ordinary differential equation) with condition is a type of mathematical equation that describes how a variable changes over time, where the variable is dependent on its own rate of change. The condition refers to an initial value or boundary value that must be satisfied for the equation to be solved.

2. How is a first order ODE with condition different from a regular first order ODE?

A first order ODE with condition is different from a regular first order ODE because it requires an initial or boundary value to be specified in order to solve it. This condition acts as a starting point for the equation, while a regular first order ODE does not have this initial value requirement.

3. What are some examples of first order ODEs with conditions?

Some examples of first order ODEs with conditions include the exponential growth and decay equations, the logistic growth equation, and the radioactive decay equation. These equations all involve a dependent variable changing over time, and have an initial or boundary value that must be specified to solve them.

4. How do you solve a first order ODE with condition?

To solve a first order ODE with condition, you can use various methods such as separation of variables, integrating factors, or the method of undetermined coefficients. These methods involve algebraic manipulation and integration to find the solution to the equation that satisfies the given condition.

5. Why are first order ODEs with conditions important in science?

First order ODEs with conditions are important in science because they are used to model and understand natural phenomena that involve change over time. Many physical, biological, and chemical processes can be described by first order ODEs, and solving them with conditions helps us make predictions and analyze real-world scenarios.

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