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 higherorder derivative can appear in this equation. A firstorder 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, firstorder initial value problem differential equations, and their applications. We will solve a few examples for a better understanding of the concept.
1.  What is First Order Differential Equation? 
2.  Solution First Order Differential Equation 
3.  Initial Value Problem First Order Differential Equation 
4.  FAQs on First Order Differential Equation 
What is First Order Differential Equation?
A First Order Differential Equation is generally written as F(x, y, y') = 0, where y' is the firstorder 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 reallife example of the firstorder 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' = t^{2} + 1 ⇒ F(t, y, y') = t^{2} + 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 firstorder 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.
Solution First Order Differential Equation
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 firstorder differential equation y' + y P(x) = 0 and then the general solution to the nonhomogeneous 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 = x^{2}/2 + C
⇒ ln 5y + 4 = 5 (x^{2}/2 + C)
⇒ ln 5y + 4 = 5x^{2}/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 = 4x^{2}  3x.
First, we will write the given firstorder differential equation in the form y' + y P(x) = Q(x)
Dividing xy' + 3y = 4x^{2}  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} = x^{3}
Multiplying the firstorder differential equation y' + (3/x) y = 4x  3 by the I.F. = x^{3}, we have
x^{3}(y' + (3/x) y) = x^{3}(4x  3)
⇒ x^{3}y' + 3x^{2}y = 4x^{4}  3x^{3}
⇒ d(yx^{3})/dx = 4x^{4}  3x^{3}
Integrating both sides of the equation with respect to x, we get
⇒ ∫[d(yx^{3})/dx] dx = ∫(4x^{4}  3x^{3}) dx
⇒ yx^{3} = (4/5)x^{5}  (3/4)x^{4} + C
⇒ y = (4/5)x^{2}  (3/4)x + Cx^{3}
Hence, the general solution of the first order differential equation xy' + 3y = 4x^{2}  3x using the integrating factor method is y = (4/5)x^{2}  (3/4)x + Cx^{3}
Initial Value Problem First Order Differential Equation
First order initial value problem differential equation is of the form F(x, y, y') = 0 and initial value y(x_{0}) = y_{0}, where x_{0} is a fixed value of x and y_{0} is the corresponding value of y and (x_{0},y_{0}) satisfies the general solution y(x). The initial value of the differential equation helps to find the particular solution of the firstorder 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 firstorder differential equation y' = x^{2} + 1, y(1) = 4
First, we will evaluate the general solution of the given differential equation. We have dy/dx = x^{2} + 1 which can be solved by separating the variables.
dy/dx = x^{2} + 1
⇒ dy = (x^{2} + 1) dx
Integrating both sides of the equation, we get
∫ dy = ∫ (x^{2} + 1) dx
⇒ y = x^{3}/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 = 1^{3}/3 + 1 + C
⇒ C = 4  1/3  1
= 3  1/3
= 8/3
Therefore y = y = x^{3}/3 + x + 8/3 is the particular solution of the initial value problem differential equation y' = x^{2} + 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.
 Firstorder differential equations can be solved using the separation of variables, integrating factor, and variation of parameters method.
Related Topics on First Order Differential Equation
First Order Differential Equation Examples

Example 1: Solve the firstorder differential equation x^{2}y' = x^{3} + 2
Solution: To solve the given differential equation, we will separate its variables.
x^{2}y' = x^{3} + 2
⇒ y' = x + 2/x^{2}
Integrating both sides w.r.t. x
⇒ ∫(dy/dx) dx = ∫ (x + 2/x^{2}) dx
⇒ y = x^{2}/2  2/x + C
Answer: Hence the general solution is y = x^{2}/2  2/x + C

Example 2: Determine the solution of the differential equation xy(dy/dx) + y^{2} + 2y = 0 using the integrating factor method.
Solution: The given first order differential equation can be written as:
xy(dy/dx) + y^{2} + 2y = 0
⇒ dy/dx + y/x + 2/x = 0 [Dividing the differential equation by xy]
⇒ dy/dx + y/x =  2/x
On comparing the differential equation dy/dx + y/x =  2/x with dy/dx + y P(x) = Q(x), we have P(x) = 1/x, and Q(x) = 2/x. Next, find the integrating factor.
I.F. = e^{∫P(x) dx} = e^{∫(1/x) dx} = e^{ln x} = x
Multiplying the firstorder differential equation dy/dx + y/x =  2/x by the I.F. = x, we have
x(dy/dx + y/x) = x(2/x)
⇒ xdy/dx + y = 2
⇒ d(xy)/dx = 2
⇒ ∫ [d(xy)/dx]dx = ∫2 dx
⇒ xy = 2x + C
⇒ y = 2 + C/x
Answer: The general solution of xy(dy/dx) + y^{2} + 2y = 0 is y = 2 + C/x.
FAQs on First Order Differential Equation
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 higherorder derivative can appear in this equation. A firstorder 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 firstorder 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 firstorder 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 firstorder derivative and no other higherorder 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.
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