Formation of Differential Equation MCQ Quiz - Objective Question with Answer for Formation of Differential Equation - Download Free PDF

Calculation

The system of circles touching Y axis at origin will have centres on X axis. Let (a,0) be the centre of a circle. Then the radius of the circle should be a units, since the circle should touch Y axis at origin.

Equation of a circle with centre at and radius

That is,

The above equation represents the family of circles touching Y axis at origin. Here 'a' is an arbitrary constant.

In order to find the differential equation of system of circles touching Y axis at origin, eliminate the arbitrary constant from equation(1)

Differentiating equation(1) with respect to x,

or

Replacing '2a' of equation(1) with the above expression, you get

That is,

or

Hence option 2 is correct

Calculation:

Given:

Differential equation: 

y = vx, differentiate with respect to x:

Substitute y = vx and into the given differential equation:

⇒ 

Subtract v from both sides:

⇒ 

Separate the variables: 

⇒ 

Hence option 3 is correct

Calculation

y = ea sin x

⇒ log y = a sin x

⇒ ⇒ = 0

⇒ y log y = tanx 

Hence option 1 is correct

Explanation:

To solve the given differential equation xdx - ydy = 0 ,

⇒ 

This is a separable differential equation. We can separate the variables as

follows 

Now, integrate both sides

Exponentiating both sides to remove the logarithms



Where  is a constant. 

This represents the equation of a straight line passing through the origin, where

k is a constant representing the slope of the line. A family of such lines would

form the geometric figure of a hyperbola in certain coordinate transformations.

The correct answer is "Hyperbola" because the equation represents a family of

hyperbolas.

thus correct option is option 2.

Solution- 

The differential equation that incorporates both the threshold population T and carrying capacity K is 

where r represents the growth rate.

Therefore, Correct answer is Option 2).

Solution- 

The differential equation that incorporates both the threshold population T and carrying capacity K is 

where r represents the growth rate.

Therefore, Correct answer is Option 2).

Solution - Consider the family of curve tangent form an angle of \(\frac{\pi}{4}\) with the hyperbola xy = c is  ,

y = f(x) 

so,  

Now, hyperbola is xy = c ⇒ y = c/x 

Slope of the tangent to the hyperbola is y' = - 

According to the Question we have,

Therefore, Correct Option is Option 3).

Explanation:

To solve the given differential equation xdx - ydy = 0 ,

⇒ 

This is a separable differential equation. We can separate the variables as

follows 

Now, integrate both sides

Exponentiating both sides to remove the logarithms



Where  is a constant. 

This represents the equation of a straight line passing through the origin, where

k is a constant representing the slope of the line. A family of such lines would

form the geometric figure of a hyperbola in certain coordinate transformations.

The correct answer is "Hyperbola" because the equation represents a family of

hyperbolas.

thus correct option is option 2.

Calculation

The system of circles touching Y axis at origin will have centres on X axis. Let (a,0) be the centre of a circle. Then the radius of the circle should be a units, since the circle should touch Y axis at origin.

Equation of a circle with centre at and radius

That is,

The above equation represents the family of circles touching Y axis at origin. Here 'a' is an arbitrary constant.

In order to find the differential equation of system of circles touching Y axis at origin, eliminate the arbitrary constant from equation(1)

Differentiating equation(1) with respect to x,

or

Replacing '2a' of equation(1) with the above expression, you get

That is,

or

Hence option 2 is correct

Calculation:

Given:

Differential equation: 

y = vx, differentiate with respect to x:

Substitute y = vx and into the given differential equation:

⇒ 

Subtract v from both sides:

⇒ 

Separate the variables: 

⇒ 

Hence option 3 is correct

Calculation

y = ea sin x

⇒ log y = a sin x

⇒ ⇒ = 0

⇒ y log y = tanx 

Hence option 1 is correct

Calculation:

Given:

y = p cos (ax) + q sin (ax)       ….(1)

Now differentiating both sides, we get

Again differentiating both sides, we get

From equation 1st,