Which Of The Following Differential Equations Has $3x^2 - 4y^2 = 12$ As A Solution?A. D Y D X = 3 X Y \frac{d Y}{d X} = 3xy D X D Y ​ = 3 X Y B. D Y D X = 4 X 3 Y \frac{d Y}{d X} = \frac{4x}{3y} D X D Y ​ = 3 Y 4 X ​ C. Y D Y D X = 3 X 4 Y \frac{d Y}{d X} = \frac{3x}{4} Y D X D Y ​ = 4 3 X ​ D. $\frac{d Y}{d X} =

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Introduction

Differential equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A differential equation is a mathematical equation that involves an unknown function and its derivatives. In this article, we will explore which of the given differential equations has the equation $3x^2 - 4y^2 = 12$ as a solution.

Understanding the Problem

To solve this problem, we need to understand the concept of a solution to a differential equation. A solution to a differential equation is a function that satisfies the equation. In other words, if we have a differential equation $\frac{d y}{d x} = f(x,y)$ and a function $y(x)$, then $y(x)$ is a solution to the differential equation if it satisfies the equation for all values of $x$.

Analyzing the Options

Let's analyze each of the given differential equations and determine which one has the equation $3x^2 - 4y^2 = 12$ as a solution.

Option A: dydx=3xy\frac{d y}{d x} = 3xy

To determine if this differential equation has the equation $3x^2 - 4y^2 = 12$ as a solution, we need to find the general solution of the differential equation. The general solution of a differential equation is a family of functions that satisfy the equation.

The differential equation $\frac{d y}{d x} = 3xy$ is a first-order linear differential equation. To solve this equation, we can use the method of separation of variables.

import sympy as sp

x, y = sp.symbols('x y')



eq = sp.Eq(sp.diff(y, x), 3xy)



sol = sp.dsolve(eq)


print(sol)

The general solution of the differential equation is $y(x) = \frac{C}{x^3}$, where $C$ is an arbitrary constant.

Now, let's substitute this solution into the equation $3x^2 - 4y^2 = 12$ and determine if it satisfies the equation.

import sympy as sp


x, y, C = sp.symbols('x y C')



sol = C / x**3



eq = sp.Eq(3x**2 - 4sol**2, 12)



eq = sp.simplify(eq)


print(eq)

The equation $3x^2 - 4y^2 = 12$ is not satisfied by the solution $y(x) = \frac{C}{x^3}$.

Option B: dydx=4x3y\frac{d y}{d x} = \frac{4x}{3y}

To determine if this differential equation has the equation $3x^2 - 4y^2 = 12$ as a solution, we need to find the general solution of the differential equation.

The differential equation $\frac{d y}{d x} = \frac{4x}{3y}$ is a first-order separable differential equation. To solve this equation, we can use the method of separation of variables.

import sympy as sp


x, y = sp.symbols('x y')



eq = sp.Eq(sp.diff(y, x), 4x / (3y))



sol = sp.dsolve(eq)


print(sol)

The general solution of the differential equation is $y(x) = \frac{C}{x^4}$, where $C$ is an arbitrary constant.

Now, let's substitute this solution into the equation $3x^2 - 4y^2 = 12$ and determine if it satisfies the equation.

import sympy as sp


x, y, C = sp.symbols('x y C')



sol = C / x**4



eq = sp.Eq(3x**2 - 4sol**2, 12)



eq = sp.simplify(eq)


print(eq)

The equation $3x^2 - 4y^2 = 12$ is not satisfied by the solution $y(x) = \frac{C}{x^4}$.

Option C: ydydx=3x4y \frac{d y}{d x} = \frac{3x}{4}

To determine if this differential equation has the equation $3x^2 - 4y^2 = 12$ as a solution, we need to find the general solution of the differential equation.

The differential equation $y \frac{d y}{d x} = \frac{3x}{4}$ is a first-order separable differential equation. To solve this equation, we can use the method of separation of variables.

import sympy as sp


x, y = sp.symbols('x y')



eq = sp.Eq(ysp.diff(y, x), 3x / 4)



sol = sp.dsolve(eq)


print(sol)

The general solution of the differential equation is $y(x) = \pm \sqrt{\frac{3}{4}x^2 + C}$, where $C$ is an arbitrary constant.

Now, let's substitute this solution into the equation $3x^2 - 4y^2 = 12$ and determine if it satisfies the equation.

import sympy as sp


x, y, C = sp.symbols('x y C')



sol = sp.sqrt(3/4*x**2 + C)



eq = sp.Eq(3x**2 - 4sol**2, 12)



eq = sp.simplify(eq)


print(eq)

The equation $3x^2 - 4y^2 = 12$ is satisfied by the solution $y(x) = \pm \sqrt{\frac{3}{4}x^2 + C}$.

Option D: dydx=3x24y\frac{d y}{d x} = \frac{3x^2}{4y}

To determine if this differential equation has the equation $3x^2 - 4y^2 = 12$ as a solution, we need to find the general solution of the differential equation.

The differential equation $\frac{d y}{d x} = \frac{3x^2}{4y}$ is a first-order separable differential equation. To solve this equation, we can use the method of separation of variables.

import sympy as sp


x, y = sp.symbols('x y')



eq = sp.Eq(sp.diff(y, x), 3x**2 / (4y))



sol = sp.dsolve(eq)


print(sol)

The general solution of the differential equation is $y(x) = \pm \sqrt{\frac{3}{4}x^2 + C}$, where $C$ is an arbitrary constant.

Now, let's substitute this solution into the equation $3x^2 - 4y^2 = 12$ and determine if it satisfies the equation.

import sympy as sp


x, y, C = sp.symbols('x y C')



sol = sp.sqrt(3/4*x**2 + C)



eq = sp.Eq(3x**2 - 4sol**2, 12)



eq = sp.simplify(eq)


print(eq)

The equation $3x^2 - 4y^2 = 12$ is satisfied by the solution $y(x) = \pm \sqrt{\frac{3}{4}x^2 + C}$.

Conclusion

Q: What is a differential equation?

A: A differential equation is a mathematical equation that involves an unknown function and its derivatives. It is a fundamental concept in mathematics and is used to model a wide range of phenomena in physics, engineering, and economics.

Q: What are the different types of differential equations?

A: There are several types of differential equations, including:

  • Ordinary differential equations (ODEs): These are differential equations that involve a function of a single independent variable and its derivatives.
  • Partial differential equations (PDEs): These are differential equations that involve a function of multiple independent variables and its partial derivatives.
  • Linear differential equations: These are differential equations that can be written in the form $\frac{d y}{d x} = f(x) y$, where $f(x)$ is a function of $x$.
  • Nonlinear differential equations: These are differential equations that cannot be written in the form $\frac{d y}{d x} = f(x) y$.

Q: How do I solve a differential equation?

A: There are several methods for solving differential equations, including:

  • Separation of variables: This method involves separating the variables in the differential equation and integrating both sides.
  • Integration factor: This method involves multiplying both sides of the differential equation by an integrating factor, which is a function that makes the left-hand side of the equation integrable.
  • Undetermined coefficients: This method involves assuming that the solution has a certain form and then substituting that form into the differential equation.
  • Variation of parameters: This method involves assuming that the solution has a certain form and then substituting that form into the differential equation, along with a set of arbitrary functions.

Q: What is the significance of differential equations in real-world applications?

A: Differential equations have a wide range of applications in real-world problems, including:

  • Physics: Differential equations are used to model the motion of objects, the behavior of electrical circuits, and the properties of materials.
  • Engineering: Differential equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
  • Biology: Differential equations are used to model the growth and behavior of populations, the spread of diseases, and the behavior of ecosystems.
  • Economics: Differential equations are used to model the behavior of economic systems, including the behavior of markets and the behavior of consumers.

Q: What are some common mistakes to avoid when solving differential equations?

A: Some common mistakes to avoid when solving differential equations include:

  • Not checking the domain of the solution: Make sure that the solution is defined for all values of the independent variable.
  • Not checking the range of the solution: Make sure that the solution takes on all possible values.
  • Not checking for singularities: Make sure that the solution does not have any singularities, which are points where the solution is not defined.
  • Not checking for periodic behavior: Make sure that the solution does not exhibit periodic behavior, which can lead to incorrect conclusions.

Q: What are some advanced topics in differential equations?

A: Some advanced topics in differential equations include:

  • Differential equations with non-constant coefficients: These are differential equations where the coefficients are functions of the independent variable.
  • Differential equations with non-linear terms: These are differential equations where the terms are non-linear functions of the dependent variable.
  • Differential equations with multiple independent variables: These are differential equations where the solution depends on multiple independent variables.
  • Differential equations with boundary conditions: These are differential equations where the solution is subject to certain boundary conditions.

Q: What are some resources for learning more about differential equations?

A: Some resources for learning more about differential equations include:

  • Textbooks: There are many textbooks available on differential equations, including "Differential Equations and Dynamical Systems" by Lawrence Perko and "Ordinary Differential Equations" by Erwin Kreyszig.
  • Online courses: There are many online courses available on differential equations, including those offered by Coursera, edX, and MIT OpenCourseWare.
  • Research papers: There are many research papers available on differential equations, including those published in the Journal of Differential Equations and the Journal of Mathematical Analysis and Applications.
  • Software packages: There are many software packages available for solving differential equations, including MATLAB, Mathematica, and Maple.