Math Problem Statement

Choose the correct answer. (1) A first order differential equation of the form y′ = f(x) is called ········· variable. (A) Non-separable (B) Linear (2) If f(x)dy = g(y)dx, then (C) Exact (D) Separable

Solution

Let's break down both parts of the question:

(1) "A first-order differential equation of the form y=f(x)y' = f(x) is called..."

  • This equation has the form dydx=f(x)\frac{dy}{dx} = f(x), which does not involve the variable yy on the right-hand side. This is a simple form of a linear differential equation. In this case, it's called Linear because it's of the form y=f(x)y' = f(x), where the derivative of yy is a function of xx only and doesn't involve nonlinear terms.

Thus, the correct answer for (1) is: (B) Linear

(2) "If f(x)dy=g(y)dxf(x)dy = g(y)dx, then the differential equation is..."

  • This equation can be rewritten as: dyg(y)=dxf(x)\frac{dy}{g(y)} = \frac{dx}{f(x)} which is a separable differential equation because the variables xx and yy can be separated on opposite sides of

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Math Problem Analysis

Mathematical Concepts

Differential Equations
First-Order Equations
Linear Equations
Separable Equations

Formulas

y' = f(x)
f(x)dy = g(y)dx
\frac{dy}{g(y)} = \frac{dx}{f(x)}

Theorems

Separable Differential Equations
Linear Differential Equations

Suitable Grade Level

Undergraduate Mathematics