First Order Differential Equation Examples with Solution
Below are the example of problems on First Order Differential Equation.
Example 1: Solve the following separable differential equation: dy/dx = x/y2
Solution:
First, we separate the variables:
y2.dy = x.dx
Integrating both sides:
∫y2.dy = ∫x.dx
y3/3 = x2/2 + C
Example 2: Solve the following linear differential equation: dy/dx + 2xy – x = 0
Solution:
Equation in the standard form:
dy/dx + 2xy – x = 0
Now, we can use an integrating factor to solve it:
f(x) = e ∫2xdx
f(x) = [Tex]e^{x^2}[/Tex]
Multiplying both sides by the integrating factor:
[Tex]e^{x^2} \frac{dy}{dx} + 2xye^{x^2} – xe^{x^2} = 0[/Tex]
[Tex]\frac{d}{dx} (ye^{x^2}) – xe^{x^2} = 0[/Tex]
Integrating both sides:
[Tex]ye^{x^2} = \frac{x^2}{2} + C[/Tex]
Example 3: Solve the first-order differential equation x3y’ = x + 2
Solution:
x3y’ = x + 2
⇒ y’ = (x + 2)/x3
Integrating both sides w.r.t. x
⇒ ∫(dy/dx) dx = ∫ {(x + 2)/x3} dx
⇒ y = -1/x – 1/x2 + C
First Order Differential Equation
A first-order differential equation is a type of differential equation that involves derivatives of the first degree (first derivatives) of a function. It does not involve higher derivatives. It can generally be expressed in the form: dy/dx = f(x, y). Here, y is a function of x, and f(x, y) is a function that involves x and y.
It defined by an equation dy/dx = f (x, y) where x and y are two variables and f(x, y) are two functions. It is defined as a region in the xy plane. These types of equations have only the first derivative dy/dx so that the equation is of the first order and no higher-order derivatives exist.
Differential equations of first order is written as;
y’ = f (x, y)
(d/dx)y = f (x, y)
Let’s learn more about First-order Differential Equations, types, and examples of First-order Differential equations in detail below.
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