Differential Equations and its Solutions
- A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation.
- A differential equation involving derivatives of the dependent variable with respect to more than one independent variable is called a partial differential equation.
- The order of a differential equation is the order of the highest order derivative appearing in the equation.
- The degree of a differential equation is the degree of the highest order derivative, when the given differential equation can be expressed as a polynomial expression in derivatives, i.e., when differential coefficients are made free from radicals and fractions.
- Steps involved in forming a differential equation from general solution curve:
Step 1: Write the equation involving independent variable x (say), dependent variable y (say) and the arbitrary constants from given information.
Step 2: Obtain the number of arbitrary constants in step 1.
Let there be n arbitrary constants.
Step 3: Differentiate the relation in step 1 with respect to x, n
Step 4: Eliminate arbitrary constants with the help of n equations involving differential coefficients obtained in step 3 and an equation in step 1.
The equation so obtained is the desired differential equation.
- The solution of a differential equation is a relation between the variables involved in the equation, which satisfies the given differential equation.
- A solution, which is free from arbitrary constants, is called a particular solution.
- A solution, which contains as many arbitrary constants as the order of the given differential equation is called the general solution.
The order and degree of the differential equation, are respectivelyMarks:1
Equation of the curve passing through (1, 1) and satisfying the differential equation is given byMarks:1
x2 = y.
The order and degree of the differential equation whose solution is y = cx + c2 - 3c3/2 + 2, where c is a parameter, are respectively.Marks:1
1 and 4.
The differential equation of all conics whose centre lie at the origin is of orderMarks:1
Explanation:The general equation of all conics whose centre lie at the origin is ax^2 + 2hxy + by^2 = 1 Clearly, it has three arbitrary constants. So, the differential equation will be of order 3.