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We can classify differential equations according to their order.
The order of a differential equation (DE) is the order of the highest order derivative of the dependent variable with respect to the independent variable involved in that particular differential equation.
The degree of a differential equation is the highest positive power of the highest order of a derivative involved in the given polynomial differential equation.
There are many different methods of solving various types of differential equations (or DE). In this article, we will be focusing on the sol. of 1st order & 1st degree DE.
The Concept of Differential Calculus and Differentiation
Differential calculus is the branch of mathematics that deals with studying the rate of change of a quantity with respect to another. It is a subfield of calculus.
The function’s derivative describes the rate of change of a function near a chosen input value. Geometrically, it is the area of the slope of a tangent to the function at that given point in a graph.
The process to find a derivative of a function is termed differentiation and is reverse to the operation of integration.
The concept of differentiation is applicable in many quantitative processes. We can apply it to various concepts of physics, chemistry and mathematics, such as:
Finding the acceleration of a moving body by differentiating its velocity with respect to time
In Newton’s second law of motion
Finding the rate of a chemical reaction
Finding the maxima and minima of a given function
Differential Equations
Equations involving derivatives of a dependent variable with respect to an independent variable are known as differential equations. For example,
x (dy/dx) – y = 0
The above equation is differential.
Differential equations that involve derivatives of a dependent function with respect to more than a single independent variable are known as partial differential equations. For example,
(∂2u/∂t2) – c2 (∂2u/∂x2) = 0.
The Order and Degree of a Differential Equation
We can define the order of a differential equation (DE) as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in that particular differential equation. Differential equations can be classified according to their order.
Differential equations involving first-order derivatives are known as first-order DE.
The degree of a differential equation is the highest positive power of the highest order of a derivative involved in the given polynomial differential equation.
For example, for the given equations:
(dy/dx) – ex = 0,
In the above equation, both the order and degree are 1.
(d3y/dx3) + x3 (d2y/dx2) + 3 (dy/dx)3 = 0
Here, the order is 3, and the degree is 1.
Sol. of 1st Order & 1st Degree DE
Finding the solution for a differential equation with order and degree equal to 1 is termed Sol. of 1st Order & 1st Degree DE.
Any 1st order & 1st degree DE can be expressed in the given form:
(dy/dx) = f (x,y)
Or,
A(x,y) dx + B(x,y) dy = 0.
Method of Solving 1st Order 1st Degree DE
Let us look at some methods that will help us solve 1st Order & 1st Degree DE equations.
By separation of variables
We can separate the variables of an equation if we convert an equation
A(x,y) dx + B(x,y) dy = 0
to the form
M(x) dx + N(y) dy = 0.
The following differential equation gives its solution:
∫ M(x) dx + ∫ N(y) dy = C, where C is an arbitrary constant.
Example:
Solve (1 + x2)dy – xy dx = 0.
By dividing both sides with y(1 + x2) and transposing them, we get:
dy/y = (x / 1 + x2)dx.
By integrating both sides, we have:
log y = ½ log (1 + x2) + log C.
log y = log C (1 + x2)1/2
y = C (1 + x2)1/2
By homogeneous DE
The DE A(x,y) dx + B(x,y) dy = 0 is homogeneous if both A(x,y) and B(x,y) are homogeneous functions of the same degree.
Here, substituting either variable with a new variable (say, v) is possible, such as y = vx or x = vy.
By exact DE
We can obtain the exact DE by equating the total differential of some function to zero, where total differential of a function f(x,y) is given by:
df(x,y) = [∂f(x,y)/ ∂x]dx + [∂f(x,y)/ ∂y]dy = 0.
By integrating factors
If an equation is not exact DE, we can multiply it with some factors to make it one.
Example:
y dx + 2x dy = 0
The above equation is not exact DE since, ∂A/∂y ≠ ∂B/∂x.
But multiplying the DE with y gives an exact DE as below:
y2 dx + 2xy dy = 0
now , it has become exact differential
d(y2x)=0 integrating both sides we get
y2x=c :solution of DE
By linear equations of 1st order
A DE of the form dy/dx + y.P(x) = Q(x) has an integrating factor e∫P(x) dx.
The solution id as below:
y = e-∫P(x) dx [∫ Q(x)eP(x) dx dx + C]
Example:
y′−y=xex here integrating factor=e-x
y=x2/2 +c :solution of DE
Conclusion
Through this easy-to-understand and well-written article, CBSE students have come to grasp the concepts of differential calculus and differentiation, the derivative of a function, differential equations in calculus, order and degree of a DE, and Sol. of 1st Order & 1st Degree DE.
They are now familiar with the different methods of finding the sol. of 1st order & 1st degree DE, namely, by separating variables, homogeneous DE, exact DE, integrating factors, and linear equations of 1st order.
