For each of the differential equations in question, find a particular solution satisfying the given condition: x(x 2 – 1) dy/dx = 1; y = 0 when x = 0

Mathematics Part-II

Book: Mathematics Part-II

Chapter: 9. Differential Equations

Subject: Maths - Class 12^th

Q. No. 12 of Exercise 9.4

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12

For each of the differential equations in question, find a particular solution satisfying the given condition:
x(x² – 1) dy/dx = 1; y = 0 when x = 0

Separating variables,

Or,

Integrating both sides,

Now let,

Or

Now comparing the values of A,B,C

A + B + C = 0

B-C = 0

A = -1

Solving these we will get that

Now putting the values of A,B,C in ii)

Now integrating it,

Now we are given that

Or,

Now putting the value of

Then,

12

Chapter Exercises

Exercise 9.1

Exercise 9.2

Exercise 9.3

Exercise 9.4

Exercise 9.5

Exercise 9.6

Miscellaneous Exercise

1

For each of the differential equations in question, find the general solution:

2

For each of the differential equations in question, find the general solution:

3

For each of the differential equations in question, find the general solution:

4

For each of the differential equations in question, find the general solution:
sec²x tan y dx + sec²y tan x dy

5

For each of the differential equations in question, find the general solution:
(e^x + e^–x)dy – (e^x – e^–x)dx = 0

6

For each of the differential equations in question, find the general solution:

7

For each of the differential equations in question, find the general solution:
y log y dx – x dy = 0

8

For each of the differential equations in question, find the general solution:

9

For each of the differential equations in question, find the general solution:

10

For each of the differential equations in question, find the general solution:
e^xtan y dx + 1(1 – e^x)sec²y dy = 0

11

For each of the differential equations in question, find a particular solution satisfying the given condition:
(x³ + x² + x + 1) dy/dx = 2x² + x, y = 1 when x = 0

12

For each of the differential equations in question, find a particular solution satisfying the given condition:
x(x² – 1) dy/dx = 1; y = 0 when x = 0

13

For each of the differential equations in question, find a particular solution satisfying the given condition:
cos dy/dx = a; y = 2 when x = 0

14

For each of the differential equations in question, find a particular solution satisfying the given condition:
dy/dx = y tan x; y = 1 when x = 0

15

Find the equation of a curve passing through the point (0, 0) and whose differential equation is y′ = e^x sin x

16

For the differential equation find the solution curve passing through the point (1, –1).

17

Find the equation of a curve passing through the point (0, –2) given that at any point (x, y) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.

18

At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (– 4, –3). Find the equation of the curve given that it passes through (–2, 1).

19

The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.

20

In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs 100 double itself in 10 years (loge2 = 0.6931).

21

In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e^0.5 = 1.648).

22

In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?

23

The general solution of the differential equation is