How do you use the Quadratic Formula to solve First Order ODE?

In summary, to solve First Order ODE using the Quadratic Formula, you can treat the constant t^2 + k as any other constant and bring it over to create a quadratic equation. From there, you can use the Quadratic Formula to express y in terms of t.
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Homework Statement



How do you use the Quadratic Formula to solve First Order ODE?

For example, I am given this integration (see attachment at the bottom).

Homework Equations


The Attempt at a Solution



I integrated both sides but I do not know where to go from there (see attachment at the bottom). My goal is to express y in terms of t. Furthermore, my book says I should use the Quadratic Formula to figure it out. I know what the Quadratic Formula is, but I do not know how to apply it in this situation. Can anyone give me any hints? Thanks.
 

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  • #2
Treat the t^2 + k as any other constant. Bring it over and you have a quadratic equation like any other.
 
  • #3
Sethric said:
Treat the t^2 + k as any other constant. Bring it over and you have a quadratic equation like any other.

I see. Thanks.
 

Related to How do you use the Quadratic Formula to solve First Order ODE?

1. How do you use the Quadratic Formula to solve First Order ODE?

The Quadratic Formula can be used to solve a First Order Ordinary Differential Equation (ODE) by converting it into a quadratic equation in terms of the dependent variable. The general form of a First Order ODE is dy/dx = f(x,y). To solve this using the Quadratic Formula, we need to rearrange the equation to the form of ax^2 + bx + c = 0, where a = 1, b = f(x,y), and c = -dy/dx. Then, we can use the Quadratic Formula, x = (-b ± √(b^2 - 4ac)) / 2a, to find the values of x that satisfy the equation.

2. What are the steps to using the Quadratic Formula to solve First Order ODE?

To use the Quadratic Formula to solve a First Order ODE, follow these steps:

  1. Rearrange the equation to the form ax^2 + bx + c = 0, where a = 1, b = f(x,y), and c = -dy/dx.
  2. Identify the values of a, b, and c.
  3. Plug in the values to the Quadratic Formula, x = (-b ± √(b^2 - 4ac)) / 2a.
  4. Solve for the values of x.
  5. Substitute the values of x back into the original equation to find the corresponding values of y.

3. Can the Quadratic Formula be used to solve any First Order ODE?

No, the Quadratic Formula can only be used to solve First Order ODEs that can be converted into quadratic equations in terms of the dependent variable. If the First Order ODE cannot be rearranged into this form, then the Quadratic Formula cannot be used to solve it.

4. Are there any limitations to using the Quadratic Formula to solve First Order ODEs?

Yes, there are a few limitations to using the Quadratic Formula to solve First Order ODEs. First, the ODE must be in the form of dy/dx = f(x,y), where f(x,y) is a function of both x and y. Second, the ODE must be a linear equation, meaning that the dependent variable y cannot be raised to a power or multiplied by itself. Third, the ODE must be autonomous, meaning that it does not explicitly depend on the independent variable x.

5. What are the benefits of using the Quadratic Formula to solve First Order ODEs?

The Quadratic Formula provides a straightforward and systematic method for solving First Order ODEs. It also allows for the solution to be expressed in terms of the independent variable, which can be useful for further analysis and interpretation. Additionally, the Quadratic Formula can be used to find both real and complex solutions to the ODE, providing a more complete understanding of the problem at hand.

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