First order differential equation with substitution

In summary, the problem involves using substitution with v = y^-2 to solve for y'. After dividing by y^3 and isolating y', the next step is to find dy/dt using the chain rule and then substitute it back into the equation to solve for v'. Ultimately, an integrating factor must be used to arrive at the final solution.
  • #1
kaitamasaki
20
0

Homework Statement



t^2 y' + 4ty - y^3 = 0

Homework Equations



Hint was given in the question: substitute with v = y^-2

The Attempt at a Solution



Dividing by t^2 and isolating y':
t^2 y' = y^3 - 4ty
y' = y^3 / t^2 - 4y/t

dv/dt = 0
y = v^(-1/2)
dy/dt = (-1/2)v^(-3/2) v'

so y' = dy/dt = (-1/2)v^(-3/2) v' = v^(-3/2) - 4t(v^(-1/2))

But after playing with algebra I cannot separate v and t,
I end up with;
dv/dt - 8v/t = -2/t^2
Where v' = dv/dt

How should I have approached this problem?
 
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  • #2
hi kaitamasaki! :smile:

(try using the X2 icon just above the Reply box :wink:)
kaitamasaki said:
Dividing by t^2 …

why t2 ? :confused:

try dividing by y3 :smile:
 
  • #3
Oh gee can't believe I missed that
Had to use integrating factor after
 
Last edited:

1. What is a first order differential equation with substitution?

A first order differential equation with substitution is a type of mathematical equation that involves an unknown function and its derivative. The substitution method involves substituting the dependent variable with a new variable in order to simplify the equation and make it easier to solve.

2. How do you solve a first order differential equation with substitution?

To solve a first order differential equation with substitution, you need to follow these steps:

  1. Identify the dependent and independent variables.
  2. Substitute the dependent variable with a new variable.
  3. Differentiate the substituted equation with respect to the independent variable.
  4. Substitute the original equation and the differentiated equation into each other.
  5. Solve the resulting equation for the new variable.
  6. Substitute the new variable back into the original equation to find the solution for the dependent variable.

3. What is the purpose of using substitution in solving differential equations?

The purpose of using substitution in solving differential equations is to simplify the equation and make it easier to solve. This allows us to find the solution to a more complex equation by using a simpler equation.

4. Can a first order differential equation with substitution have multiple solutions?

Yes, a first order differential equation with substitution can have multiple solutions. This is because the substitution method involves replacing the dependent variable with a new variable, and there can be multiple ways to do this. Therefore, the solution obtained may not be unique.

5. What are some real-world applications of first order differential equations with substitution?

First order differential equations with substitution have various real-world applications, such as in physics, engineering, and economics. They are used to model and analyze dynamic systems, such as population growth, chemical reactions, and electric circuits. They are also used in the field of control systems to design and optimize systems for desired performance.

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