Solution of Differential Equations MCQ Quiz - Objective Question with Answer for Solution of Differential Equations - Download Free PDF

Calculation

Given equation:

Divide by x:

Let y = vx, then

Substitute in the equation:

⇒ 

⇒ 

⇒ 

Integrate both sides:

⇒ 

⇒ 

Remove the logarithms:

⇒ 

Substitute v = y/x:

⇒ 

∴ The general solution is .

Hence option 2 is correct

Concept:

The solution of first order differential equation  is given by y × IF = , where IF = Integrating Factor = 

Calculation:

Given, 

⇒ 

∴ I.F. = 

∴ 

Let tan^–1 x = z ⇒ 

∴ ye^z = 

⇒ 

⇒ 

Now, y(1) = 0 

⇒ 0 = 

⇒ C = 

∴ 

⇒ y(0) = 

∴ The value of y(0) is .

The correct answer is Option 2.

Calculation

Let

⇒ 

⇒ 

Integrating both sides:-

⇒ 

⇒ 

Hence, option 1 is correct

Calculation

.....

Take,

The given equation becomes

Hence option 2 is correct

Concept:

Calculation:

Given: dy = (1 + y²) dx

Integrating both sides, we get

⇒ y = tan (x + c)

∴ The solution of the given differential equation is y = tan (x + c).

Given:

x = 1

x2 + y2 + z2 = xy + yz + zx

Calculations:

x² + y2 + z2 - xy - yz - zx = 0

⇒(1/2)[(x - y)² + (y - z)² + (z - x)²] = 0

⇒x = y , y = z and z = x

But x = y = z = 1

so, 

= {10(1)⁴ + 5(1)⁴ + 7(1)⁴}/{13(1)²(1)²+ 6(1)²(1)² + 3(1)²(1)²}

= 22/22

= 1

Hence, the required value is 1.

Calculation:

Given: 

On integrating both sides, we get

⇒ y = xe^a + c

Concept:

Calculation:

Given: 

Integrating both sides, we get

Given:

x +  = 3

Concept Used:

Simple calculations is used

Calculations:

⇒ x +  = 3

On multiplying 2 on both sides, we get

⇒ 2x +  = 6  .................(1)

Now, On cubing both sides,

⇒ 

⇒ 

⇒ 

⇒ 

⇒   ..............from (1)

⇒ 

⇒ 

⇒ Hence, The value of the above equation is 180

Concept: 

Calculation: 

Given :  

⇒  

Integrating both sides, we get 

⇒  

⇒ 

⇒   [∵ 2c = C]

⇒ 

The correct option is 2 . 

Concept:

Some useful formulas are:

If log x = z then we can write x = e^z

Calculation:

Rearranging the equation and integrating we get,

⇒

⇒, c = constant of integration

⇒ log(3x + 8) = 3(t + c)

⇒ 3x + 8 = e^3(t+c) 

⇒ 3x = e^3(t+c) - 8

∴ 

Concept:

Calculation:

Given: dy =  dx 

⇒  

Integrating both sides, we get

⇒  

⇒  = x + c 

⇒ y = sin ( x + c ) . 

The correct option is 2.

Concept:

Calculation:

Given: 

Integrating both sides, we get

Calculation:

Given: ​

⇒ ydy = (x + 1) dx

Integrating both sides, we get

⇒ ∫ ydy = ∫ (x + 1) dx

⇒ 

⇒ y² = x² + 2x + 2c

∴ y2 - x2 - 2x - c = 0

Please note: c is constant here, so 2c can be also considered as a constant.