About simple first order differential equation

In summary, the given differential equation dy/dx = (3x^2+4x+2)/2(y-1) has a solution of y = 1±√(x^3+2x^x+2x) for x≥0, as specified by the initial condition y(0)=1. This is because when x=0, the function is not defined and the positive sign solution satisfies the initial condition.
  • #1
jwqwerty
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Homework Statement


dy/dx = (3x^2+4x+2)/2(y-1) , y(0)=1


Homework Equations





The Attempt at a Solution


I get the answer and the steps are shown:

2(y-1)dy=(3x^2+4x+2)dx and integrate both sides
y^2-2y=x^3+2x^2+2x+c
By initial condition, c=-1 and by solving for y,
y = 1±√(x^3+2x^x+2x) (x>0)

But i do not understand why x≥0 isn't the answer. If it is because of y≠1 (x=0 when y=1) by the equation above, then why does initial condition hold true?
 
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  • #2


Thank you for your question. I can understand why you are confused about the answer to this problem. Let me explain it to you in more detail.

First of all, let's take a look at the initial condition given in the problem: y(0)=1. This means that when x=0, the value of y is 1. Plugging this into the equation, we get:

dy/dx = (3(0)^2+4(0)+2)/2(1-1) = 2/0

We can see that this is undefined, which means that the function is not defined at x=0. This is why we need to specify that x≥0 in the answer, because at x=0, the function is not defined.

Now, let's take a look at the solution you have provided. You have correctly integrated both sides of the equation and solved for y. However, when you take the square root, you need to remember that there are two possible solutions, one with the positive sign and one with the negative sign. This is why you have the ± in your solution. The initial condition y(0)=1 only holds true for the solution with the positive sign, which is why the answer is x≥0 and not x>0.

I hope this clarifies your confusion. Please let me know if you have any further questions. Good luck with your studies!
 

Related to About simple first order differential equation

1. What is a simple first order differential equation?

A simple first order differential equation is a mathematical expression that describes the relationship between a function and its derivative. It only involves one independent variable and its derivative, and it can be solved using basic algebraic and calculus methods.

2. What is the general form of a simple first order differential equation?

The general form of a simple first order differential equation is dy/dx = f(x), where y is the dependent variable, x is the independent variable, and f(x) is a function of x.

3. How do you solve a simple first order differential equation?

To solve a simple first order differential equation, you can use various methods such as separation of variables, integrating factor, or substitution. These methods involve manipulating the equation to isolate the dependent and independent variables and then integrating to find the solution.

4. What are the applications of simple first order differential equations?

Simple first order differential equations are used to model a wide range of physical phenomena, such as population growth, radioactive decay, and chemical reactions. They also have applications in engineering, economics, and other fields.

5. Can a simple first order differential equation have multiple solutions?

Yes, a simple first order differential equation can have multiple solutions. This is because the general solution of a first order differential equation contains a constant of integration, which can take on different values and result in different solutions.

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