Problem 1 - Calculus

Question

Determine which of the following differential equations are separable. Do not solve the equations. (a) \(y^{\prime}=y\) (b) \(y^{\prime}=x+y\) (c) \(y^{\prime}=x y\) (d) \(y^{\prime}=\sin (x+y)\) (e) \(y^{\prime}-x y=0\) (f) \( y^{\prime}=y / x\) (g) \(y^{\prime}=\ln (x y)\) (h) \(y^{\prime}=(\sin x)(\cos y)\) (i) \(y^{\prime}=(\sin x)(\cos x y)\) (j) \( y^{\prime}=x / y\) (k) \( y^{\prime}=2 x\) (l) \(y^{\prime}=(x+y) /(x+2 y)\)

Step-by-Step Solution

Answer

Equations (a), (c), (e), (f), (h), (j), (k) are separable.

Step 1: Define Separable Differential Equations

A differential equation is separable if it can be expressed in the form \( \frac{dy}{dx} = g(x)h(y) \), where \( g(x) \) is a function of \( x \) only, and \( h(y) \) is a function of \( y \) only.

Step 2: Identify Separable Equations

Begin checking each differential equation to see if it can be written in the form \( \frac{dy}{dx} = g(x)h(y) \).(a) \( y' = y \) is separable: \( y' = 1 \cdot y \), so \( g(x)=1, h(y)=y \).(b) \( y' = x + y \) is not separable because it can't be split into \( g(x) \) and \( h(y) \).(c) \( y' = xy \) is separable: \( y' = x \cdot y \), so \( g(x)=x, h(y)=y \).(d) \( y' = \sin(x+y) \) is not separable due to the mixed term inside the sine function.(e) \( y' - xy = 0 \) can be rewritten as \( y' = xy \): is separable, \( g(x)=x, h(y)=y \).(f) \( y' = \frac{y}{x} \) is separable: \( y' = \frac{1}{x} \cdot y \), so \( g(x)= \frac{1}{x}, h(y)=y \).(g) \( y' = \ln(xy) \) is not separable due to the logarithmic mixed term.(h) \( y' = \sin(x)\cos(y) \) is separable: \( y' = (\sin(x)) \cdot (\cos(y)) \), so \( g(x)=\sin(x), h(y)=\cos(y) \).(i) \( y' = \sin(x)\cos(xy) \) is not separable because of the mixed term in the cosine function.(j) \( y' = \frac{x}{y} \) is separable: \( y' = x \cdot \frac{1}{y} \), so \( g(x)=x, h(y)=\frac{1}{y} \).(k) \( y' = 2x \) is separable: \( y' = 2x \cdot 1 \), so \( g(x)=2x, h(y)=1 \).(l) \( y' = \frac{x+y}{x+2y} \) is not separable because the function cannot be expressed as \( g(x)h(y) \).

Key Concepts

Differential Equations

Differential equations are a fundamental concept in calculus, describing how a particular quantity changes in relation to another. Specifically, they are mathematical equations that involve functions and their derivatives. Differential equations can describe a variety of phenomena such as how populations change over time or how heat transfers in different environments.

There are several types of differential equations, but in this context, we focus on separable differential equations. Recognizing whether a differential equation is separable is essential for solving it using specific techniques. The primary consideration is whether both variables, typically represented as functions of time or space, can be separated and thus manipulated independently.

By identifying if a differential equation is separable, one can determine the appropriate solution strategies, streamlining the problem-solving process in calculus. Understanding the nature of differential equations allows mathematicians and scientists to model dynamic systems effectively.

Solution Techniques in Calculus

In the study of calculus, various solution techniques are developed to tackle different forms of mathematical problems, differential equations being one of them. For a differential equation to be solved, understanding its type is crucial.

**Separable Differential Equations** are one such type and are characterized by their ability to be decomposed into a product of two functions, one involving only the independent variable and one involving only the dependent variable. The equation can thus be expressed in the form \( \frac{dy}{dx} = g(x)h(y) \). This form allows the integration of each side independently, a powerful technique since it can simplify complex differential equations into a more manageable form.

The method involves the following steps:

• Rearrange the equation so all terms involving the dependent variable are on one side, and all terms with the independent variable are on the other.
• Integrate both sides separately, obtaining expressions for both \(y\) and \(x\).
• Solve for the dependent variable if necessary, obtaining the general solution.

Identifying separable equations is a pivotal part of calculus, as this recognition leads to efficient solving techniques.

Mathematical Functions

Mathematical functions are the foundation upon which calculus is built, and they are crucial when dealing with differential equations. Functions express relationships between variables and are essential for understanding the behavior of mathematical models.

In differential equations, functions often relate two quantities, with their derivatives providing insight into the rate of change or the slope of the function at any point. These derivatives are central to calculus and understanding how functions change over time or space.

In the context of separable differential equations, recognizing how functions can be split into parts is vital. These parts are expressed as products of functions \(g(x)\) and \(h(y)\), reflecting the independent and dependent variable aspects of the equation respectively.

• Functions of the form \(g(x)\) represent the independent variable aspect.
• Functions of the form \(h(y)\) emphasize the dependent variable's behavior.

Understanding these functions and how they interact in differential equations helps in applying the correct calculus techniques. This is foundational for solving many real-world problems and achieving a deeper comprehension of mathematical concepts.