B.2.2.2 solve DE ....separate variables

• MHB
• karush
In summary: So, what Romsek is saying is that you can "multiply" both sides by dx, and then use the properties of derivatives to get y'= y(x+h) and dy= f'(x)dx.
karush
Gold Member
MHB
#2

$y'= \dfrac{x^2}{y(1+x^3)}$

Separate y dy =\dfrac{x^2}{(1+x^3)...

ok i tried to get the book ans but someahere derailed
why is =c in the answer

$$\displaystyle y^\prime = \dfrac{x^2}{y(1+x^3)}\\~\\ y dy = \dfrac{x^2}{1+x^3}dx\\~\\ \dfrac 1 2 y^2 = \dfrac{1}{3} \log \left(x^3+1\right) + C\\~\\ y^2 = \dfrac 2 3 \log(x^3+1) + C~\text{(note the constant absorbs multiplication by other constants)}\\~\\ y = \pm \left(\dfrac 2 3 \log(x^3+1) + C\right)^{1/2} = \pm \sqrt{\dfrac 2 3} \left(\log(x^3+1)+C\right)^{1/2}\\~\\ \text{C is in the answer because there is a whole family of curves parameterized by C that are a solution to this diff eq.}\\~\\ \text{You can select a specific curve by specifying initial conditions.}$$

ok i don't see how you can just
put dx in separations

great help tho appreciate the steps

karush said:
ok i don't see how you can just
put dx in separations
Huh? What step are you not understanding?

-Dan

the second step the dy was already there the dx wasn't

$$\displaystyle y^\prime = \dfrac{dy}{dx}$$

so just "multiply" both sides by $$\displaystyle dx$$

Have you read on how to solve separable diff eqs?

Before you take Differential Equations you need to know Calculus really well! I think you need more practice in Calculus.

You say " the dy was already there the dx wasn't ".
Strictly speaking neither was in the original equation- it had y'. But I am sure you understand that y' and dy/dx are just different notations for the derivative of y with respect to x.

Romsek said "just multiply both sides by dx". Now, Strictly speaking "dy/dx" is NOT a fraction, it is rather the limit of fractions: $\lim_{h\to 0}\frac{y(x+h)- y(x)}{h}$. But it can be "treated like a fraction". To make use of that we define (typically in Calculus 3) the "differentials" dy and dx separately from dy/dx such that dy= y' dx. That allows us to treat dy/dx as if it were a fraction and say if dy/dx= f'(x) then dy= f'(x)dx.

1. What is the concept of "separating variables" in solving differential equations?

The concept of separating variables in solving differential equations involves isolating the dependent and independent variables on opposite sides of the equation to make it easier to solve. This is done by rearranging the equation and integrating both sides separately.

2. How do you know when to use the method of separating variables in solving a differential equation?

The method of separating variables is typically used when the differential equation is in the form of dy/dx = f(x)g(y), where f(x) and g(y) are functions of the independent and dependent variables, respectively. This method can also be used when the equation can be manipulated into this form.

3. What are the steps involved in solving a differential equation by separating variables?

The steps involved in solving a differential equation by separating variables are as follows:
1. Rearrange the equation so that all terms with the dependent variable are on one side and all terms with the independent variable are on the other side.
2. Integrate both sides of the equation separately.
3. Add a constant of integration to one side of the equation.
4. Solve for the dependent variable to get the general solution.
5. If initial conditions are given, use them to find the particular solution.

4. What are some common mistakes to avoid when using the method of separating variables to solve a differential equation?

Some common mistakes to avoid when using the method of separating variables include:
- Forgetting to add a constant of integration
- Incorrectly integrating one or both sides of the equation
- Not isolating the dependent and independent variables on opposite sides of the equation
- Forgetting to substitute the initial conditions into the particular solution
- Making algebraic errors during the solving process.

5. Can the method of separating variables be applied to all types of differential equations?

No, the method of separating variables can only be applied to certain types of differential equations, specifically those in the form of dy/dx = f(x)g(y). Other types of differential equations may require different methods, such as substitution or using an integrating factor.

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