Differential Equations

Differential Equations are those types of equations that have some Derivatives of certain Functions. The derivatives can either be ordinary derivatives or partial derivatives. If there are only ordinary derivatives in the equation then, the equation is defined as the ordinary type of differential equation and if the equation has all its terms as partial derivative then, such type of equation is called as partial differential equation. For example, let’s consider an equation: 

a(dy/dx) = f(x,y).....(equation 1)

The above example shows the simplest form of the differential equation. Some other examples can be taken as:

ax’’ + bx’ + cx = f(y)......(equation 2)

a² d²u/ dx² = du/ dx.....(equation 3)

sin(y) d²u/ dx² = (1-2y) dy/dx + y³e-2y _…..(equation 4)

x⁽⁴⁾ + 4y’’’ – 5y = sin(t).....(equation 5)

d³u/ d²xdt = 1 – du/ dx.....(equation 6)

Now, let's talk about the order of the differential equations. The order of any differential equation is the highest derivative present in the equation. Here, in above examples, equation 1 is first order equation, equations 2 and 3 are the type of second order differential equations, equation 6 is the third order differential equation and equation 5 is fourth order differential equation. The types of differential equations are based on the order of these equations. Proceeding further, let’s discuss about types of Differential Equations, we have linear differential equations, ordinary differential equation, and partial differential equations. The linear types of differential equations are equations that can be expressed in the form of:

a₁yⁿ(t) + a₂y^(n-1)(t) + a₃y^(n-2)(t)………+ y₂(t) + y(t)  = f(t)

The coefficients of equation a₀, a₁, a₂ etc. can be zero or nonzero Numbers, linear or non-linear function, and constant or non-constant Functions. If an equation cannot be written in the form of linear equation then it is called as non-linear type of equation.

The above equations 3, 4, and 6 are the partial type of equations and ordinary equations are equations 1, 2, and 5. Now, solve few problems related to differential equations.

For instance: dy/dx = x³ - 5, 

So, we have to integrate this equation with respect to x to find the solution of the equation and also add a variable in the solution (k). This composition is the general solution for the differential equation. Let us see an example:

dy/ dx = x³ – 5x + 5

Now, this equation can be written in differential form as:

Y = ∫ (x3 – 5x + 5) dx

Y = x⁴ /4 – 5x²/2 + 5x + k

This is the general solution for the differential equation.

Differential equations are most commonly used in real life. To find the velocity of a freely falling object when there is only the gravity and resistance of the air in the environment. They are also used to find the exponential growth of any quantity.