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Explain in Detail :General Solution of 1st Order Differential Equation
A first-order differential equation is an equation that describes the change in a function over time in terms of its first derivative. The general solution of a first-order differential equation is a function that solves the equation and includes all the arbitrary constants of integration.
There are a few methods that can be used to find the general solution of a first-order differential equation. One method is to use the separation of variables technique. This technique involves dividing the equation into two parts: one that contains only the derivative and one that contains only the function. This technique can then be used to solve each of the two parts separately.
Another method that can be used to find the general solution of a first-order differential equation is the integration technique. This technique involves integrating the equation to find the function. Once the function is found, the arbitrary constants of integration can be determined.
First-Order Differential Equation
A differential equation is an equation that relates a function and one or more of its derivatives. A first-order differential equation is a differential equation that has one derivative.
The most common type of first-order differential equation is the linear equation. A linear equation is an equation in which the sum of the powers of the variables is the same on each side of the equation.
The general form of a linear first-order differential equation is:
y'(x) = a(x)y(x) + b(x)
Where:
y'(x) is the derivative of y(x)
a(x) is a constant
b(x) is a constant
First Order Linear Differential Equation
A first order linear differential equation is an equation in which the first derivative of a function is linear in the function’s argument. A first order linear differential equation typically has the form:
,
where is a function and is its first derivative.
How to Solve 1st Order Differential Equations?
There are a few steps in solving 1st order differential equations:
1. Assume a particular solution to the equation.
2. Use algebra to solve for the particular solution.
3. Check the solution by plugging it back into the differential equation.
Solving 1st order differential equations can be tedious but the two approaches mentioned below can ease things out.
Solution using Separation of Variables
This is the most common technique for solving 1st order linear differential equations. The idea is to divide the equation into two parts – one that contains only the derivatives and one that contains only the constants. This can be done by multiplying both sides of the equation by a common factor if needed.
After the equation has been divided, the two parts can be solved separately.
Here is an example:
The equation is:
The equation can be divided by y to get:
The equation can be divided by x to get:
The two parts can now be solved separately.
The derivative part can be solved using the Product Rule:
The constant part can be solved using the Quadratic Formula:
The final solution is:
Solution using Integration
This approach is useful when the differential equation cannot be solved using separation of variables.
The first step is to integrate both sides of the equation. After integration, the equation will look like an algebraic equation.
Next, solve the algebraic equation.
Here is an example:
The equation is:
The equation can be integrated to get:
After integration, the equation becomes an algebraic equation.
Next, solve the algebraic equation.
The solution is:
