Differential Equations Formula
Differential Equations is the higherorder differential and is applicable in situations involving complex changes. Differential equation formulas find numerous applications in biology, physics, quantum mechanics, engineering. Differential Equations Formula includes the concepts and formulas from differentiation and integrations.
What is the Differential Equations Formula?
The differential equations formulas includes higher order differentials such as d^{n}y/dx^{n}. There are four important formulas for differential equations. The formulas to find the order, degree of the differential equation, and to work across homogeneous, linear differential equations.
Formula 1
Order of a Differential Equation is the order of the highest order differential equation. Similar to the general equations with a variable x, here we have the differential dy/dx, which is written with varying degrees as exponents. Here 'n' is the order of this differential equation.
d^{n}y/dx^{n} + ......d^{3}y/dx^{3 }+ d^{2}y/dx^{2}+ dy/dx + k = 0
Formula 2
The degree of a Differential Equation is the degree of the highest order derivative. Here the order of the differential equation is 'n' and the degree of the differential equation is 'a'.
(d^{n}y/dx^{n})^{a} + .....(.d^{3}y/dx3)^{4} + (d^{2}y/dx^{2})^{2}+ (dy/dx) + k = 0
Formula 3
Homogenous Differential Equation refers to an equation in which the substitution of x, y with λx, λy, can be manipulated to get λ^{n} common for the entire expression.
f(λx, λy) = λ^{n}f(x, y)
Formula 4
Linear Differential Equation is similar to a normal equation, but with a variation of the variables. Rather than x and y as variables, here we have dy/dx and y as variables. This is a linear differential equation in y and P and Q are the constants or expressions in 'x'.
dy/dx + Py = Q is the differential equation.
General solution of the differential equation is \(y = e^{\int P.dx}.\int(Q.e^{\int P.dx}).dx + C\)
Let us check out a few solved examples to understand more about differential equation formulas.
Solved Examples on Differential Equations Formula

Example 1: Find the order and degree of the differential equation (d^{3}y/dx^{3})^{2 }+ (d^{2}y/dx^{2})^{3 }+ (dy/dx)^{2} + 5 = 0.
Solution:
The given differential equation is (d^{3}y/dx^{3})^{2} + (d^{2}y/dx^{2})^{3} + (dy/dx)^{2} + 5 = 0
Order: It is the order of the highest derivative in the equation, and it is 3
Degree: It is the power of the highest derivative in the equation, and it is 2.
Answer: Hence the order is 3 and the degree is 2. 
Example 2: Find the general solution of the differential equation dy/dx = 4/x.
Solution:
The given differential equation is dy/dx = 4/x
To find the general solution we need to separate the x and y terms and then integrate on both sides.
dy/dx = 4/x
dy = 4/x.dx
Integrate on both sides.
\(\int \).dy = \(\int \) 4/x.dx
y = 4logx + C
Answer: Therefore the general solution is y = 4 logx + c.