Differential Eq. Last Step Solution Separating Variables

In summary, to solve the differential equation y' = (3x^2 - 1) / (3+2y), we can use the method of completing the square and taking the square root or the quadratic formula to obtain the general solution expressed in explicit form. It is important to be cautious of any missing terms in the process of solving the equation.
  • #1
knowLittle
312
3
Solve each differential equation. Express the general solution in explicit form.

y' = (3x^2 -1) / (3+2y)

So, I will skip many steps, because they are easy. However, I am stuck in one of the last ones.
y^2 +3y = x^3- x +C

y(y+3)= x^3 - x +C

I have seen the solution for y, but I don't understand how it is derived. Can someone help?
 
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  • #2
Try completing the square on the left hand side, and then taking the square root of both sides of the equation.
 
  • #3
y(y+ 3)= y^2+ 3y= x^3- x+ C

y^2+ 3y+ (x- C)= 0.

Solve that quadratic equation using the quadratic formula.
 
  • #4
HallsofIvy said:
y(y+ 3)= y^2+ 3y= x^3- x+ C

y^2+ 3y+ (x- C)= 0.

Solve that quadratic equation using the quadratic formula.

Hello HallsofIvy, How did you get rid of x^3 ?
 
  • #5
knowLittle said:
Hello HallsofIvy, How did you get rid of x^3 ?

He appears to inadvertently left it out. The method HallsofIvy suggested and the method I suggested are mathematically equivalent.

Chet
 
  • #6
knowLittle said:
Hello HallsofIvy, How did you get rid of x^3 ?

Chestermiller said:
He appears to inadvertently left it out.
Yes.
Chestermiller said:
The method HallsofIvy suggested and the method I suggested are mathematically equivalent.
Move all of the terms to the left side, and you'll have a quadratic in y. It's probably simpler at this point to just use the Quadratic Formula to solve for y, which will have two parts separated by ±, as is usually the case.

If there is an initial condition, it might lead you to choose one or the other of the two values.
 
  • #7
knowLittle said:
Hello HallsofIvy, How did you get rid of x^3 ?
I waved my magic wand and uttered a spell of "Carelessness"!
 

FAQ: Differential Eq. Last Step Solution Separating Variables

What is the concept of Separating Variables in Differential Equations?

Separating variables is a method used to solve differential equations by separating the independent and dependent variables on opposite sides of the equation. This allows us to solve for the dependent variable by integrating both sides.

How do you know when to use Separating Variables in solving a Differential Equation?

Separating variables is typically used when the differential equation is in the form of dy/dx = f(x)g(y). In this form, the dependent and independent variables can easily be separated by dividing both sides by g(y) and then integrating.

What are the steps to solving a Differential Equation using Separating Variables?

The steps to solving a differential equation using separating variables are as follows:

  1. Rewrite the differential equation in the form dy/dx = f(x)g(y)
  2. Divide both sides by g(y) to separate the variables
  3. Integrate both sides with respect to x and add a constant of integration, C
  4. Solve for y by isolating it on one side of the equation

What are the common mistakes to avoid when using Separating Variables to solve a Differential Equation?

Some common mistakes to avoid when using separating variables to solve a differential equation include:

  • Forgetting to divide both sides by g(y) before integrating
  • Not adding a constant of integration, C, when integrating
  • Forgetting to solve for y after integrating
  • Incorrectly differentiating or integrating a function

Can Separating Variables be used to solve all types of Differential Equations?

No, separating variables can only be used to solve differential equations in the form of dy/dx = f(x)g(y). Other techniques, such as using the integrating factor or substitution, may be needed to solve other types of differential equations.

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