For the differential equations , find the general solution:\(\frac {dy}{dx}+y=1, \ \ (y≠1)\)
Solution and Explanation
The given differential equation is:
\(\frac {dy}{dx}+y=1\)
\(⇒dy+y dx=dx\)
\(⇒dy=(1-y)dx\)
Separating the variables, we get:
\(\frac {dy}{1-y}=dx\)
Now, integrating both sides, we get:
\(∫\frac {dy}{1-y}=∫dx\)
\(⇒log\ (1-y)=x+log\ C\)
\(⇒-log\ C-log\ (1-y)=x\)
\(⇒log\ C(1-y)=-x\)
\(⇒C(1-y)=e^{-x}\)
\(⇒1-y=\frac {1}{C}e^{-x}\)
\(⇒y=1+Ae^{-x }\) \((where \ A=-\frac 1C)\)
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