# 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 times.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.

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• Q1

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1.

##### Explanation:

• Q2

The order and degree of the differential equation, are respectively

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2, 2.

##### Explanation:
• Q3

Equation of the curve passing through (1, 1) and satisfying the differential equation is given by

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x2 = y.

##### Explanation:

• Q4

The order and degree of the differential equation whose solution is y = cx + c2 - 3c3/2 + 2, where c is a parameter, are respectively.

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1 and 4.

##### Explanation:

• Q5

The differential equation of all conics whose centre lie at the origin is of order

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3.

##### 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.