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

    Marks:1
    Answer:

    1.

    Explanation:

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  • Q2

    The order and degree of the differential equation, are respectively

    Marks:1
    Answer:

    2, 2.

    Explanation:
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  • Q3

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

    Marks:1
    Answer:

    x2 = y.

    Explanation:

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  • 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.

    Marks:1
    Answer:

    1 and 4.

    Explanation:

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  • Q5

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

    Marks:1
    Answer:

    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.
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