Differential Equations - Exemplar Solutions

CBSE Class–12 Mathematics

NCERT Exemplar
Chapter - 9
DIFFERENTIAL EQUATIONS - Short Answer Questions

1. Find the differential equation of the family of curves .

Sol. Given .

Differentiating on both sides with respect to x

Thus,

2. Find the general solution of the differential equation

Sol.Given

(Using formula logA +logB=logAB in RHS and removing log on both sides)

3. Given that and . Find the value of when .

Sol. Given

Substituting we get i.e., $l o g e = 1$

Therefore, .

Now, substituting x = 1 in the above, we get .

4. Solve the differential equation

Sol. The equation is of the type , which is a linear differential equation.

Now I.F. (elogf(x)=f(x))

Therefore, solution of the given differential equation is

Hence,

5. Find the differential equation of the family of lines through the origin.

Sol. Let be the family of lines through origin. Therefore, .

Eliminating m, we get

6. Find the differential equation of all non-horizontal lines in a plane.

Sol. The general equation of all non-horizontal lines in a plane is , where .

differentiating both sides w.r.t. y on both sides,we get

Again, differentiating both sides w.r.t. y, we get

7. Find the equation of a curve whose tangent at any point on it, different from origin, has slope

Sol. Given

Integrating on both sides, we get

$\int \frac{d y}{y} = \int (1 + \frac{1}{x}) d x$

Long Answer Questions

8. Find the equation of a curve which is passing through the point (1, 1).If the perpendicular distance of the origin from the normal at any point of the curve is equal to the distance of P form the .

Sol. Let the equation of normal at

Therefore, the length of perpendicular from origin to (1) is

Also, distance between . Thus, we get

Case I:

Integrating both sides, we get Substituting we get .

Therefore, is the equation of curve (not possible, so rejected).

Case II: . Substituting we get

$x \frac{d v}{d x} = \frac{v^{2} - 1}{2 v} - v$

Integrating both sides, we get

Substituting we get .

Therefore, is the required equation.

9. Find the equation of a curve passing through .If the slope of the tangent to the curve at any point is

Sol. According to the given condition

slope of tangent,m=dy/dx

This is a homogeneous differential equation. Substituting we get

$\int \sec^{2} v d v = \int - \frac{d x}{x}$

Substituting , we get Thus, we get

which is the required equation.

10. Solve and

Sol. Given equation can be written as

Dividing both sides by we get

Integrating both sides, we get

Substituting we get

therefore, $\tan (\frac{y}{2 x}) = \frac{- 1}{2 x^{2}} + \frac{3}{2}$ is the required solution.

11. State the type of the differential equation for the equation. and solve it.

Sol. Given equation can be written as

$\frac{d y}{d x} = \frac{\sqrt{x^{2} + y^{2}}}{x} \frac{+ y}{}$

Clearly RHS of (1) is a homogeneous function of degree zero. Therefore, the given equation is a homogeneous differential equation. Substituting we get form (1)

Integrating both sides of (2), we get

Objective Questions

Choose the correct answer form the given four options in each of the Examples 12 to 21.

12. The degree of the differential equation is

(A) 1

(B) 2

(D) 4

Sol. The correct answer is (B).

13. The degree of the differential equation is

(A) 1

(B) 2

(D) Not defined

Sol. Correct answer is (D). The given differential equation is not a polynomial equation in terms of its derivatives, so its degree is not defined.

14. The order and degree of the differential equation respectively, are

(A) 1, 2

(B) 2, 2

(D) 4, 2

Sol. Correct answer is (C).

15. The order of the differential equation of all circles of given radius is:

(A) 1

(B) 2

(D) 4

Sol. Correct answer is (B). Let the equation of given family be It has two arbitrary constants h and k. Therefore, the order of the given differential equation will be 2.

16. The solution of the differential equation represents a family of

(A) straight lines

(B) circles

(D) ellipses

Sol. Correct answer is (C). Given equation can be written as

which represents the family of parabolas.

17. The integrating factor of the differential equation

(A)

(B)

(C)

(D)

Sol. Correct answer is (B). Given equation can be written as

Therefore, I.F.

18. A solution of the differential equation is

(A)

(B)

(C)

(D)

Sol. Correct answer is (C).

19. Which of the following is not a homogeneous function of .

(A)

(B)

(C)

(D)

Sol. Correct answer is (D).

20. Solution of the differential equation is

(A)

(B)

(C)

(D)

Sol. Correct answer is (C). From the given equation, we get giving .

21. The solution of the differential equation is

(A)

(B)

(C)

(D)

Sol. Correct answer (D). Therefore, the solution is

22. Fill in the blanks of the following:

(i) Order of the differential equation representing the family of parabolas is ____________.

Sol. One; a is the only arbitrary constant.

(ii) The degree of the differential equation is _______________.

Sol. Two; since the degree of the highest order derivative is two.

(iii) The number of arbitrary constants in a particular solution of the differential equation is _______________.

Sol. Zero; any particular solution of a differential equation has no arbitrary constant.

(iv) is a homogeneous function of degree _________________.

Sol. Zero.

(v) An appropriate substitution to solve the differential equation $\frac{d y}{d x} = \frac{x^{2} \log (\frac{x}{y}) + x^{2}}{x y \log (\frac{x}{y})}$ is ________.