For the differential equations, find the general solution:\(\frac{dy}{dx}=\frac{1-cosx}{1+cosx}\)
Solution and Explanation
The given differential equation is:
\(\frac{dy}{dx}=\frac{1-cosx}{1+cosx}\)
\(⇒\frac{dy}{dx}={2sin^2\frac{x}{2}}{2cos^2\frac{x}{2}}=tan^2\frac{x}{2}\)
\(⇒\frac{dy}{dx}=(sec^2\frac{x}{2}-1)\)
separating the variables,we get:
\(dy=(sec^2\frac{x}{2}-1)dx\)
Now,integrating both sides of this equation,we get:
\(∫dy=∫(sec^2\frac{x}{2}-1)dx=∫sec^2\frac{x}{2}dx-∫dx\)
\(⇒y=2tan\frac{x}{2}-x+C\)
This is the required general solution of the given differential equation.
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Concepts Used:
Differential Equations
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
Orders of a Differential Equation
First Order Differential Equation
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
Second-Order Differential Equation
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations