An equation that contain one or more derivatives or differentials is called the differential equation.
Order and Degree of Differential equation
Order: The order of highest derivatives present in the equation determines the order of D.E.
Degree: The maximum power of highest derivatives determine the degree of the D.E.
Note: First equation has been cleared from fractional and radical signs in the dependent variables and its derivatives.
Methods of differential equations
There are five methods such as follows
- Variable Separable Form
- Reducible To Variable Separation Form
- Homogeneous D.E.
- Exact D.E.
- Linear D.E.
Variable Separable Form
By simple adjustment if it is possible to write all the term containing x along with dx and the term containing y along with dy, then the equation is said to be in variable separable form.
If f(x) dx = g(y) dy
Then the direct integration of such an equation gives the general solution of the equation.
i.e. ∫ f(x) dx = ∫ g(y) dy is a general solution. is a general solution.
Reducible to Variable Separable Form
Sometime in differential equations the variable cannot be separated easily so we have to put proper substitution and then we will easily separate the variables and we will used variable separable form.
Homogeneous Differential Equation.
If the sum of degree of variables in each term in equation is same then the equation is called homogeneous equation.
Then substitute y = t x
Exact Differential Equation.
An equation is in the form of M dx +N dy = 0 then it is said to be exact differential equation.
If equation in the form of
dy / dx +Py = Q , where P and Q are constant or function of x only.
If equation is in the form of
dx / dy + Px = Q , where P and Q are constant or function of y only.
Then this equation is Linear Differential Equation
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