First Order Differential Equation

A first order differential equation is a differential equation where the maximum order of a derivative is one and no other higher-order derivative can appear in this equation. A first-order differential equation is generally of the form F(x, y, y') = 0, where y is a dependent variable and x is an independent variable and y' appears explicitly in the differential equation. It can also be written as F(t, f(t), f'(t)) = 0, where f(t) is the solution of the differential equation. A linear first order differential equation is a differential equation with a derivative of order one and the degree of the equation is also one.

In this article, we will explore the concept of first order differential equations, ways to find their solutions, first-order initial value problem differential equations, and their applications. We will solve a few examples for a better understanding of the concept.

A First Order Differential Equation is generally written as F(x, y, y') = 0, where y' is the first-order derivative and appears explicitly in the equation, x is an independent variable and y is a function of x. We say that the function f(t) is a solution of the first order differential equation F(t, f(t), f'(t)) = 0, for all values of t. A real-life example of the first-order differential equation is Newton's law of cooling equation given by, y' = k(M - y) and it can be expressed as F(t, y, y') = k(M - y) - y'. Let us see some other examples of the differential equations of first order:

• y' = t² + 1 ⇒ F(t, y, y') = t² + 1 - y'
• y' = 2(25 - y) ⇒ F(t, y, y') = 2(25 - y) - y'
• mv'(t) = -mg ⇒ F(t, v, v') = -mg - mv'(t)

Linear First Order Differential Equation

A linear first order differential equation is of the form y' + y P(x) = Q(x) or dy/dx + y P(x) = Q(x), where y, P, Q are functions of x, and y' is the first-order derivative of y. Such differential equations have a degree of the derivative equal to one, that is why it is called the linear first order differential equation. We can solve such equations using the method of integrating factors.

Now, we can solve first order differential equations using different methods such as separating the variables, integrating factors method, variation of parameters, etc. We can determine a particular solution p(x) and a general solution g(x) corresponding to the homogeneous first-order differential equation y' + y P(x) = 0 and then the general solution to the non-homogeneous first order differential equation y' + y P(x) = Q(x) is given by y(x) = p(x) + g(x). Let us solve a few examples using different methods to understand the application of each method.

Separable First Order Differential Equation

The general form of the separable first order differential equation is dy/dx = f(y).g(x). Here we can separate the variables on the two sides of the equation, i.e., dy/dx = f(y).g(x) can also be written as dy/f(y) = g(x) dx by separating the variables and then we can solve the equation by integration. Let us consider an example of a first order differential equation and find its solution by the separation of variables method.

Example: Consider differential equation dy/dx = (5y + 4)x.

Keeping the variable y on the LHS and x on the RHS of the equation, we have

dy/dx = (5y + 4)x

⇒ dy/(5y + 4) = xdx

Now, integrating both sides of the equation, we have

∫dy/(5y + 4) = ∫ x dx

⇒ (1/5) ln |5y + 4| = x²/2 + C

⇒ ln |5y + 4| = 5 (x²/2 + C)

⇒ ln |5y + 4| = 5x²/2 + 5C

⇒ 5y + 4 = e^{5x2/2 + 5C}

⇒ y = (1/5) [e^{5x2/2 + 5C} - 4]

⇒ y = (1/5)e^5x2/2 e^5C - 4/5

⇒ y = Ke^5x2/2 - 4/5, where K = (1/5)e^5C

Hence, y = Ke^5x2/2 - 4/5 is the general solution of the first order differential equation dy/dx = (5y + 4)x.

Solving First Order Differential Equation Using Integrating Factors

A first order differential equation of the form dy/dx + y P(x) = Q(x) can be solved using the integrating factors method. We can follow the given steps to find the general solution of the differential equation:

• Step 1: Simplify the first order differential equation and express it as dy/dx + y P(x) = Q(x)
• Step 2: Determine the integrating factor given by, I.F. = e^{∫P(x) dx}
• Step 3: Multiply the differential equation dy/dx + y P(x) = Q(x) by the I.F. to obtain d(y × I.F.)/dx = Q(x) × I.F.
• Step 4: Now, integrate both sides of the equation d(y × I.F.)/dx = Q(x) × I.F. to get the general solution.

Let us solve a first order differential equation using the integrating factor method to understand its application.

Example: Consider the differential equation xy' + 3y = 4x² - 3x.

First, we will write the given first-order differential equation in the form y' + y P(x) = Q(x)

Dividing xy' + 3y = 4x² - 3x by x, we have

y' + (3/x) y = 4x - 3

Here P(x) = 3/x and Q(x) = 4x - 3

Now, the integrating factor I.F. = e^{∫P(x) dx} = e^{∫(3/x) dx} = e^{3ln x} = x³

Multiplying the first-order differential equation y' + (3/x) y = 4x - 3 by the I.F. = x³, we have

x³(y' + (3/x) y) = x³(4x - 3)

⇒ x³y' + 3x²y = 4x⁴ - 3x³

⇒ d(yx³)/dx = 4x⁴ - 3x³

Integrating both sides of the equation with respect to x, we get

⇒ ∫[d(yx³)/dx] dx = ∫(4x⁴ - 3x³) dx

⇒ yx³ = (4/5)x⁵ - (3/4)x⁴ + C

⇒ y = (4/5)x² - (3/4)x + Cx^-3

Hence, the general solution of the first order differential equation xy' + 3y = 4x² - 3x using the integrating factor method is y = (4/5)x² - (3/4)x + Cx^-3

First order initial value problem differential equation is of the form F(x, y, y') = 0 and initial value y(x₀) = y₀, where x₀ is a fixed value of x and y₀ is the corresponding value of y and (x₀,y₀) satisfies the general solution y(x). The initial value of the differential equation helps to find the particular solution of the first-order differential equation. Let us consider an example to see how to determine the solution of a differential equation using the initial value.

Example: Consider the first-order differential equation y' = x² + 1, y(1) = 4

First, we will evaluate the general solution of the given differential equation. We have dy/dx = x² + 1 which can be solved by separating the variables.

dy/dx = x² + 1

⇒ dy = (x² + 1) dx

Integrating both sides of the equation, we get

∫ dy = ∫ (x² + 1) dx

⇒ y = x³/3 + x + C, which is the general solution of the given first order differential equation.

We have y(1) = 4. Substitute this in the general solution to determine the value of C, and hence the particular solution.

4 = 1³/3 + 1 + C

⇒ C = 4 - 1/3 - 1

= 3 - 1/3

= 8/3

Therefore y = y = x³/3 + x + 8/3 is the particular solution of the initial value problem differential equation y' = x² + 1, y(1) = 4.

Important Notes on First Order Differential Equation

• Some of the important applications of the first order differential equation are in Newton's law of cooling, growth and decay models, and electric circuits.
• First-order differential equations can be solved using the separation of variables, integrating factor, and variation of parameters method.

Related Topics on First Order Differential Equation

• Rules of Differentiation
• Product Rule Formula
• Chain Rule Formula

What is First Order Differential Equation in Calculus?

A first order differential equation is a differential equation where the maximum order of a derivative is one and no other higher-order derivative can appear in this equation. A first-order differential equation is generally of the form F(x, y, y') = 0.

What is a Linear First Order Differential Equation?

A linear first order differential equation is of the form y' + y P(x) = Q(x) or dy/dx + y P(x) = Q(x), where y, P, Q are functions of x, and y' is the first-order derivative of y. Such differential equations have a degree of the derivative equal to one, that is why it is called the linear first order differential equation.

What is a Homogeneous First Order Differential Equation?

A first order differential equation M(x, y) dx + N(x, y) dy = 0 is said to be homogeneous if both M(x, y) and N(x, y) are homogeneous. We can write a homogeneous linear first-order differential equation is of the form y' + p(x)y = 0.

How to Find the General Solution of First Order Differential Equation?

The general solution of the first order differential equation can be evaluated using the separation of variables method or using the integrating factor method.

How to Find the Integrating Factor of First Order Differential Equation?

The integrating factor of the first order differential equation dy/dx + y P(x) = Q(x) can be calculated using the formula I.F. = e^{∫P(x) dx}.

What is First Order First Degree Differential Equation?

First Order First Degree Differential Equation is a differential equation involving the first-order derivative and no other higher-order derivative can appear in this equation and the degree of the derivative is one.

How Do You Solve First Order Differential Equation?

The first order differential equation can be solved using various methods such as separation of variables, integrating factor or variation of parameters.