Math Problem Statement

Which of the following differential equations is a variable separable type of differential equation? The options are: A. 4(x^2 + 1) dx + 2x tan y dx = (x^2 + 1) sec^2 y dy, B. dy - y^3 e^(2x) sin x dx - y dx = 0, C. [(y - 2y^3) e^(xy) dx + x e^(xy) dy] + (6y^2 - 2) y dy = 0, D. (xy + x) dx = (x^2 y^2 + x^2 + y^2 + 1) dy.

Solution

The question is asking which of the given differential equations is of the variable separable type. In variable separable equations, the differential equation can be rewritten such that all terms involving xx are on one side and all terms involving yy are on the other side. Let's examine each equation to check for separability.

Option A:

4(x2+1)dx+2xtanydx=(x2+1)sec2ydy4(x^2 + 1) dx + 2x \tan y \, dx = (x^2 + 1) \sec^2 y \, dy This equation has mixed terms involving both xx and yy on both sides of the equation, and it cannot be easily separated into functions of xx and yy alone. Not separable.

Option B:

dyy3e2xsinxdxydx=0dy - y^3 e^{2x} \sin x \, dx - y \, dx = 0 This equation can be rewritten as: dy=y3e2xsinxdx+ydxdy = y^3 e^{2x} \sin x \, dx + y \, dx This is not separable because yy and xx are multiplied together, and it cannot be easily rearranged into a separable form. Not separable.

Option C:

[(y2y3)exydx+xexydy]+(6y22)ydy=0\left[ (y - 2y^3) e^{xy} dx + x e^{xy} dy \right] + (6y^2 - 2) y \, dy = 0 This equation has complex terms involving both xx and yy inside exponentials and products, making it hard to separate into terms solely involving xx and yy. Not separable.

Option D:

(xy+x)dx=(x2y2+x2+y2+1)dy(xy + x) dx = (x^2 y^2 + x^2 + y^2 + 1) dy This equation can be separated because the terms involving xx and yy can be divided and moved to opposite sides: (xy+x)(x2y2+x2+y2+1)=dydx\frac{(xy + x)}{(x^2 y^2 + x^2 + y^2 + 1)} = \frac{dy}{dx} While it appears complicated, it is possible to manipulate this equation into a separable form. Separable.

Answer:

Option D is the variable separable differential equation.


Would you like further details on how to separate the variables for Option D or explanations of any specific steps? Here are some related questions you might find helpful:

  1. What are the steps to solve a separable differential equation?
  2. How can nonlinear terms affect the separability of a differential equation?
  3. Can all first-order differential equations be made separable?
  4. What are common strategies to simplify complex separable equations?
  5. How can exponential and trigonometric functions impact the separation process?

Tip: When trying to identify separable equations, always look for a way to rewrite the equation as a product of functions of xx and yy only, each on opposite sides of the equation.

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Math Problem Analysis

Mathematical Concepts

Differential Equations
Variable Separation
Algebraic Manipulation

Formulas

General separable form: dy/dx = g(x)h(y)

Theorems

Variable separable differential equations theorem

Suitable Grade Level

Grades 11-12, College Level