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QUESTION: 1

Which of the following differential equations is known as Clairaut’s equation?

Solution:

QUESTION: 2

The general solution of the differential equation (y - px) (p - 1) = p is given by

Solution:

**Proof:** The given differential equation is

(y - px) (p - i) = p ...... (i)

......(ii)

Differentiating (ii) w.r. t. x, we get

⇒ p = c .......(iii)

Eliminating p from equations (ii) and (iii),

QUESTION: 3

The general solution of i he differential equation sin(y - px) = p is given by

Solution:

QUESTION: 4

The differential equation e^{3x}(p - 1) + p^{3} e^{2}y = 0 can be reduced to Clairaut’s form by means of the substitution

Solution:

**Proof : **The given differential equation is

Substituting from (ii) and (iii) in (i), we get

or v = up + p^{2} ......(iv)

Thus equation (iv) is in Ciairaut's form.

QUESTION: 5

Which of the following differential equations cannot be transformed to the Clairaut’s form by means of the substitution x^{2} = u and y^{2} = v ?

Solution:

The differential equation in statement (d) can not be reduced to Ciairaut’s form by the transformation

x^{2} = u and y^{2 }= v ....(i)

Proof : The transform ation (i) yields

.....(ii)

Substitute in the given differential equation from (i) and (ii) and find that the transformed equation is not in Ciairaut’s form.

QUESTION: 6

The differential equation x^{2}p^{2} + yp(2x + y) + y^{2} = 0 can be reduced to the Clairaut’s form by means of which of the following substitutions?

Solution:

Hint : The transform aticn xy = v and y = u

implies that

The given equation will be reduced to

v = up + p^{2}

which is in the Clairaut's form.

QUESTION: 7

Which of the following statements is incorrect regarding the singular solutions of the differential equations?

Solution:

QUESTION: 8

To obtain the singular solution of the differential equation in Clairaut’s form it is necessary to calculate

Solution:

If the given differential equation is in Ciairaut's form, then any one of the p and c discriminants can be used for finding the singular solution of the differential equation.

**Remark :** The equation in Clairaut’s form is

y = xp +f(p) ...(i)

Its general solution is given by

y = cx + f(d) ...(ii)

Differentiating (ii) partially w.r. t. c,

0 = x + f'(c) ...(iii)

The singular solution which is the envelope of (ii) can now be obtained by eliminating c from (ii) and (iii).

**Alternately : **Differentiate (i) partially w.r. t. p,

0 = x + f'(P) ...(iv)

The singular solution can now be obtained by eliminating p from (i) and (iv).

QUESTION: 9

The singular solution (s) of the differential equation 4xp^{2} = (3x - a)^{2} is/arc given by

Solution:

QUESTION: 10

The differential equation p^{2} + y^{2} = 1 has how many singular solutions?

Solution:

**Proof : **The given differential equation is P + = 1 ...(i)

...(iii)

⇒ y = sin (x + c) ...(iii)

Differentiating (iii) w.r.t. c, we get 0 = cos (x + c)

⇒ x + c - π/2 ...(iv)

Substituting for x + c from (iv) in (iii), one singular solution is given by

This gives another singular solution as y = -1.

∴ There are 2 singular solutions

Note : complete case - 2.

QUESTION: 11

The differential equation p^{2}(1 - x^{2}) = 1 - y^{2} has how many singular solutions?

Solution:

Obtain all the four singular solutions.

QUESTION: 12

The singular solution(s) of the differential equation is/are given by

Solution:

**Proof :** The given differential equation is

......(i)

...(ii)

Differentiating (ii)partially with respect to c, we get

Substituting for c from (iii) in (i), one singular solution is given by

or x = 0

Similarly, if we rewrite (i) as

then another singular solution will be y = 0

∴ x = 0 arid y = 0

are both singular solutions.

QUESTION: 13

The singular solution(s) of the differential equation 8ap^{3} = 27y is/are given by

Solution:

QUESTION: 14

The general solution of the differential equation (a^{2} - x^{2})p^{2} + 2xyp + (b^{2} - y^{2}) = 0 represents a family of straight lines whose envelope is the

Solution:

QUESTION: 15

The Euler’s method provides the solution of which of the following firsi order differential equations

Solution:

Euler's method provided the solution of all I he first order differential equations of the form

QUESTION: 16

The Euler’s method for solving a differential equation of the type dy/dx = f(x, y) provides

Solution:

**Euler’s method**.

This method provides the solution of the differential equations of the form

in the form of a set of tabulated values. This method is very slow and to obtain reasonable accuracy with Euler's method, h should be taken very small. This method provid es an approximate solution of equation (i) in general.

QUESTION: 17

The solution of the differential equation dy/dx = f(x, y) through a given point (x_{0}, y_{0}) can be written as y = F(x). The Euler's method determines discrete points on the solution curve y = F(x). Let (x_{1} , y_{1},) be first point calculated and let (y')_{0} denote the value of dy/dx at (x_{0}, y_{0}) then

Solution:

Proof : Integrating the differential equation

QUESTION: 18

Let (x_{2}, y_{2}) be the second point calculated by the Euler’s method on the solution curve y = F(x). Then

Solution:

QUESTION: 19

Let the solution of the differential equation dy/dx = x + y pass through (0, 1) and let (0.05, y_{1}) be the first point calculated on the solution curve by the Euler’s method. Then y_{1} is equal to

Solution:

QUESTION: 20

Consider the differential equation

This will be differential equation with constant coefficients if

Solution:

If the coefficients a_{0}, a_{1} and a_{2} are constants in the differential eauation.

then it will be adifferential equation with constant coefficients.

Remark : There is no conditon on f(x) and it may be a constant (zero or non-zero) or a function of x.

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