Ans. Given: Equation of the family of curves   ……….(i) 

Here there are two arbitrary constants a and b,  therefore we will differentiate both sides two times w.r.t. x,

  

  ……….(ii)

Again differentiating w.r.t. ,

=>  

Multiplying both sides by 

, which the required differential equation.

Ans. Given: Equation of the family of curves  ……….(i) 

Here there are two arbitrary constants a and b, therefore we will differentiate both sides two times w.r.t. x,

  

 ……….(ii)

Again differentiating w.r.t. x,

  ……….(iii)

Putting this value of -a from eq (i) in eq. (ii), we get

=> 

 

 

Ans. Given: Equation of the family of curves  ……….(i) 

Here there are two arbitrary constants a and b, therefore we will differentiate both sides two times w.r.t. x,

   ……….(ii)

Again differentiating w.r.t. x

=>d2ydx2=9ae3x+4be−2x=>d2ydx2=9ae3x+4be−2x ……….(iii)

Multiplying eq. (i) by 3 and subtracting eq. (ii) from it, we get

 ……….(iv)

Again multiplying eq. (ii) by 3 and subtracting it from eq. (iii), we get

 ……….(v)

Now, eq. (v) + 2 eq. (iv) gives,

 

 , which is required differential equation.

Ans. Given: Equation of the family of curves  ……….(i) 

Here there are two arbitrary constants a and b, therefore we will differentiate both sides two times w.r.t. x

  

 

  [By eq. (i)]……….(ii)

Again differentiating w.r.t. x,

=>   ……….(iii)

Now from eq. (ii),

Putting this value of  in eq. (iii),

=> 

 

 

Ans. Given: Equation of the family of curves  ……….(i) 

Here there are two arbitrary constants a and b, therefore we will differentiate both sides two times w.r.t. x

  

 

   [By eq. (i)]……….(ii)

Again differentiating w.r.t. ,

=>  

 

       [By eq. (i)]

 

Ans. It is clear that if a circle touches y-axis at the origin must have its centre on x-axis, because x-axis being at right angles to y-axis is the normal or line of radius of the circle. 

Therefore, the centre of the circle is (r,0) where r is the radius of the circle.

  Equation of the required circle is 

  

   ……….(i)

Here r is the only arbitrary constant.

  differentiating w.r.t. x, we get

  ……….(ii)

Putting the value of 2r from eq. (ii) in eq. (i), we get

 

 

 

 , which is the required differential equation.

Ans. We know that equation of parabolas having vertex at origin and axis along positive axis is  

  ……….(i)

Here a is the only arbitrary constant. Therefore differentiating w.r.t. x, we get

 ……….(ii)

  [From eq. (i)]

 

 

 , which is the required differential equation.

Ans. We know that equation of ellipse having foci on y-axis i.e., vertical ellipse with major axis as y-axis is   ………..(i) 

Here there are two arbitrary constants a and b, therefore we will differentiate both sides two times w.r.t. x,

  

 

  ……….(ii)

Again differentiating w.r.t. ,

=>   ……….(iii)

Putting the value of  from eq. (iii), in eq. (ii), we get

  

 

Ans. We know that equation of hyperbolas having foci on x-axis and centre at origin is 

 ……….(i)

Here there are two arbitrary constants a and b, therefore we will differentiate both sides two times w.r.t. x

  

 

 ……….(ii)

Again differentiating w.r.t. ,

=>  

  ……….(iii)

Dividing eq. (iii) by eq. (ii), we get

 

  , which is required differential equation.

Ans. We know that on y-axis,  

  Centre of the circle on y- axis is .

 Equation of the circle having centre on y-axis an radius  unit is

   ……….(i)

Here  is the only arbitrary constant, therefore we will differentiate only once.

 

 

 

 

Putting this value of  in eq. (i), we get

=> 

 

 

 

Ans. Given:  ……….(i) 

  

  

   [From eq. (i)]

 

Therefore, option (B) is correct.

Ans. Given:  

  

 

On putting these values in the given option, we get the correct answer in option (C).

L.H.S. of differential equation of option (C) = 

= 

= 

= R.H.S. of option (C)

Therefore, option (C) is correct.