NCERT Solutions for class 12th Mathematics Chapter 9 Differential Equations

Exercise 9.1

Determinate order and degree (if defined) of the differential equations given in Questions 1 to 10

Question 11. The degree of the differential equation

Question 12. The order of the differential equation

Exercise 9.2

Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation

Question 11. The number of arbitrary constants in the general solution of a differential equation of fourth order are :

(a) 0 (b) 2 (c) 3 (d) 4

Sol. (d) The general solution of a differential equation of fourth order has 4arbitrary constants.

Question 12. The number of arbitrary constants in the particular solution of a differential equation of third order are :

(a) 3 (b) 2 (c) 1 (d) 0

Sol. (d) Number of arbitrary constants = 0

Exercise 9.3

In each of the following, Q. 1 to 5 form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

Note: In case of two arbitrary constants, i.e; a & b in the above question, In order to find the differential equation, we need to remove these arbitrary constants by differentiating. The given equation twice. However, if the given equation has ‘x’ bitrary constants then in order to find the differential equation we need to differentiate the given equation ‘x’ times.

Question 6. Form the differential equation of the family of circles touching the y axis at origin.

Question 7. Form the differential equation of the family of parabolas having vertex at origin and axis along positive yâ€“axis

Question 8. Form the differential equation of family of ellipses having foci on y-axis and centre at origin.

Question 9. Form the differential equation of the family of hyperbolas having foci on x axis and centre at the origin

Question 10. Form the differential equation of the family of circles having centre on y axis and radius 3 units

Question 11. Which of the following differential equation has y c e c e = + 1 2 x x â€“ as the general solution ?

Question 12. Which of the following differential equation has y = x as one of its particular solution ?

Exercise 9.4

For each of the following D.E in Q. 1 to 10 find the general solution:

Note: In case of variable separable form, we separate x and y variables and then integrate to find the general solution.

Find a particular solution satisfying the given condition for the following differential questions in Q.11 to 14.

Note: In case of particular solution, we first find the solution as done in the previous questions. Then we find the value of integrating constant by putting the values of x and y (given) in the solution. In this way integrating constant is removed and particular solution is found.

Question 15. Find the equation of the curve passing through the point (0,0) and whose differential equation yt = ex sin x

Question 16. For the differential equation xydy/dx= (x + 2) (y + 2) find the solution curve passing through the point (1, â€“ 1)

Question 17. Find the equation of a curve passing through the point (0, â€“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.

Question 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 (â€“ 4, â€“ 3) find the equation of the curve given that it passes through (â€“ 2, 1).

Question 19. The volume of a 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

Question 20. In a bank principal increases continuously at the rate of r% per year. Find the value of r if `100 double itself in 10 years (loge 2 = 0.6931).

Question 21. In a bank, principal increases continuously at the rate of 5% per year. An amount of ` 1000 is deposited with this bank, how much will it worth after 10 years (e0.5 = 1Ã—648)

Question 22. In a culture the bacteria count is 1,00,000. The number is increased by10% in 2 hours. In how many house will the count reach 2,00,000 if the rate of growth of bacteria is proportional to the number present.

Question 23. The general solution of a differential equation

Exercise 9.5

Show that the given differential equation is homogeneous and solve each of them in Questions 1 to 10

For each of the following differential equations in Q 11 ot 15 find the particular solution satisfying the given condition :

Question 16. A homogeneous equation of the form dx/dy = h(x/y) can be solved by making the substitution. (a) y = vx (b) v = yx (c) x = vy (d) x = v

Sol. (c) Option x = vy or v = x/y is correct

Question 17. Which of the following is a homogeneous differential equation?

Exercise 9.6

Find the general solution of the following differential equations in Q.1 to 12

For each of the following Questions 13 to 15, find a particular solution, satisfying the given condition :

For each of the following Questions 13 to 15, find a particular solution, satisfying the given condition :

Question 16. Find the equation of the curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.

Question 17. Find the equation of the curve passing through the point (0,2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5

Question 18. The integrating factor of the differential equation

Question 19. The integrating factor of the differential equation

Miscellaneous Exercise

Question 1. For each of the differential equations given below, indicate its order and degree (if defined)

Sol. (i) Order = 2, degree = 1. (ii) Order = 1, degree = 3. (iii) Order = 4, degree not defined

Question 2. For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

Question 3. Form the differential equation representing the family of curves given by(x â€“ a)2 + 2y2 = a2, where a is an arbitrary constant

Question 4. Prove that x2 â€“ y2 = c (x2 + y2)2 is the general solution of differential equation(x3 â€“ 3x y2) dx = (y3 â€“ 3x2y) dy, where c is a parameter.

Question 5. Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

Question 6. Find the general solution of the differential equation

Question 7. Show that the general solution of the differential equation

Question 8. Find the equation of the curve passing through the point 0,(0,pi/4) whose differential equation is sin x cos y dx + cos x sin y dy = 0.

Question 9. Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0

Question 10.

Question 11. Find a particular solution of the differential equation (x â€“ y) (dx + dy) = dx â€“dy, given that y = â€“ 1, when x = 0

Question 12. Solve the differential equation:

Question 13. Find a particular solution of the differential equation dy/dx+ y cot x = 4x cosecx (x not= 0), given that y = 0 when x =pi/ 2

Question 14. Find a particular solution of the differential equation (x + 1)dy/dx= 2eâ€“y â€“ 1,given that y = 0 when x = 0

Question 15. The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20, 000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?

Question 16. The general solution of the differential equation

Question 17. The general solution of a differential equation of the type

Question 18. The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is

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