Differential Equations, Separable, Simplification of answer

In summary, differential equations are mathematical equations used to describe how a quantity changes over time. A separable differential equation is one that can be written as two separate functions, making it easier to solve. Simplifying the answer when solving a differential equation helps with interpretation and comparison. Not all differential equations can be solved using separation of variables, and the solution can be checked by substitution or plotting.
  • #1
Destroxia
204
7

Homework Statement


I believe I have solved this differential equation, yet do not know how the book arrived at it's answer...

Solve the differential equation in its explicit solution form.

question.png


The answer the book gives is...

answer.png


Homework Equations



Separable Differential Equation

The Attempt at a Solution



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

(4y^3)dy = (x^3+x)dx

∫(4y^3)dy = ∫(x^3+x)dx [/B]

y^4 = 1/4x^4 + 1/2x^2 + c

(initial condition, y(0) = -1/sqrt(2))

(-1/sqrt(2))^(4) = 0 + 0 + c

C = -1/4
...

y^4 = 1/4x^4 + 1/2x^2 - 1/4

y = (1/4x^4 + 1/2x^2 - 1/4)^(1/4)

-----------

I've experimented with simplifying this a bit and found a few other ways to express it, but nothing like what the book has written as the answer.
 
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  • #2
Your sign on C is wrong.

##(\frac{x^4}{4}+\frac{x^2}{2} + \frac 14 ) = (\frac{x^2}{2} + \frac 12 )^2 ##
 
  • #3
RyanTAsher said:

Homework Statement


I believe I have solved this differential equation, yet do not know how the book arrived at it's answer...

Solve the differential equation in its explicit solution form.

question.png


The answer the book gives is...

answer.png


Homework Equations



Separable Differential Equation

The Attempt at a Solution



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

(4y^3)dy = (x^3+x)dx

∫(4y^3)dy = ∫(x^3+x)dx [/B]

y^4 = 1/4x^4 + 1/2x^2 + c

(initial condition, y(0) = -1/sqrt(2))

(-1/sqrt(2))^(4) = 0 + 0 + c

C = -1/4
...

y^4 = 1/4x^4 + 1/2x^2 - 1/4

y = (1/4x^4 + 1/2x^2 - 1/4)^(1/4)

-----------

I've experimented with simplifying this a bit and found a few other ways to express it, but nothing like what the book has written as the answer.
First. You made a mistake in finding C. What is (-1/sqrt(2))4 ? Fixing that will allow some factoring in the resulting expression.
 
  • #4
RUber said:
Your sign on C is wrong.

##(\frac{x^4}{4}+\frac{x^2}{2} + \frac 14 ) = (\frac{x^2}{2} + \frac 12 )^2 ##

Oh wow, I don't think I would have seen that factor regardless. Thank you though. That helped a lot.
 
  • #5
SammyS said:
First. You made a mistake in finding C. What is (-1/sqrt(2))4 ? Fixing that will allow some factoring in the resulting expression.

Thank you, I understand now. In regards to the -, out front the answer from the book, I understand that comes from the square root, but how do they determine whether to go with the - or + solution. I haven't learned intervals of validity within the book yet...
 
  • #6
The radical implies the positive. Your initial condition forces the negative choice.
 
  • #7
Before rejecting an answer, you should plug it into the diff eq and see if it works.
 

Related to Differential Equations, Separable, Simplification of answer

1. What are differential equations?

Differential equations are mathematical equations that describe how a quantity changes over time. They are often used to model natural phenomena in various scientific fields, such as physics, biology, and economics.

2. What does it mean for a differential equation to be separable?

A separable differential equation is one that can be written in the form of two separate functions multiplied together, with one function depending only on the independent variable and the other depending only on the dependent variable. This allows the equation to be solved by integrating both sides separately.

3. Why is it important to simplify the answer when solving a differential equation?

Simplifying the answer when solving a differential equation helps to make the solution more manageable and easier to interpret. It also allows for easier comparison with other solutions and can help to identify patterns or relationships between different equations.

4. Can all differential equations be solved using separation of variables?

No, not all differential equations can be solved using separation of variables. This method only applies to equations that are separable, as mentioned in the answer to the second question. Other methods, such as substitution, may be needed to solve more complex equations.

5. How can one check if the solution to a separable differential equation is correct?

One way to check the solution to a separable differential equation is to substitute the solution back into the original equation and see if it satisfies the equation. Another method is to plot the solution and compare it to the given initial conditions to ensure that it meets the required criteria.

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