Differential Equation NAT Level - 2 - Physics MCQ

# Differential Equation NAT Level - 2 - Physics MCQ

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## 10 Questions MCQ Test Topic wise Tests for IIT JAM Physics - Differential Equation NAT Level - 2

Differential Equation NAT Level - 2 for Physics 2024 is part of Topic wise Tests for IIT JAM Physics preparation. The Differential Equation NAT Level - 2 questions and answers have been prepared according to the Physics exam syllabus.The Differential Equation NAT Level - 2 MCQs are made for Physics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Differential Equation NAT Level - 2 below.
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Differential Equation NAT Level - 2 - Question 1

### Find the orthogonal trajectory of the family of curves, y2 = x3 – 2 so, find the value of y as x → ∞.

Detailed Solution for Differential Equation NAT Level - 2 - Question 1

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Differential Equation NAT Level - 2 - Question 2

### If the P.I. of the Differential Equation  is of the value form  Find the value of λ.

Detailed Solution for Differential Equation NAT Level - 2 - Question 2

This is an example of differential equation reducible to Homogeneous Equations.

Now, put  x + a = ez.
or   z = log(x + a)
Now,

⇒ λ = 1/3 = 0.333

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Differential Equation NAT Level - 2 - Question 3

### If the solution of   is of the y2 = a1x + a2xα. Find the value of  α.

Detailed Solution for Differential Equation NAT Level - 2 - Question 3

Integrating Both sides, we get

Now, it is a Bernoulli's Equation, so we put

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Differential Equation NAT Level - 2 - Question 4

If ex and x are solutions of Homogeneous equations (1 – x)y2 + xy1 – y = 2(x – 1)2e–x then the general solution y is of the form y = c1x + c2ex + α - xe-x.  Find the value of α.

Detailed Solution for Differential Equation NAT Level - 2 - Question 4

Let y1 = ex and y2 = x be two solutions of given Differential Equation. Thus,

By the method of variation of parameters

Hence, α = 0.5

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Differential Equation NAT Level - 2 - Question 5

Find the order of differential equation of all tangents to parabola y2 = x

Detailed Solution for Differential Equation NAT Level - 2 - Question 5

Equation of parabola is

y2 = x

At any point (hk) on parabola

Equation of tangent
(y – k) = m(x – h)

Hence, order of differential equation = 1.

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Differential Equation NAT Level - 2 - Question 6

If the P.I. of the differential equation (D2 – 1)y = coshxcosx is of the form α.(2 sin hx sin x - cosx).

Detailed Solution for Differential Equation NAT Level - 2 - Question 6

Combining both, we get

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Differential Equation NAT Level - 2 - Question 7

An integrating factor of the equation dy/dx + x sin 2y = x3 cos2 y is f(x). Find the value of f(0).

Detailed Solution for Differential Equation NAT Level - 2 - Question 7

Dividing complete equation by cos2y, we get.

Put tan y = t

This is Bernoulli’s equations

Now, it is a linear Differential Equation.

∴   f(x) =
f(0) = e0 = 1

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Differential Equation NAT Level - 2 - Question 8

If the PI of differential equation (D2 - 4D  + 4) y = 8x2e2x sin 2x is of the form eαx [(α'x2 + α") sin βx + β' x cos βx] then find the value of α".

Detailed Solution for Differential Equation NAT Level - 2 - Question 8

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Differential Equation NAT Level - 2 - Question 9

If the integrating factor of the differential equation (x7y2 + 3y)dx + (3x8t - x)dy = 0 is of the form xmyn then sum of values of m and n is :

Detailed Solution for Differential Equation NAT Level - 2 - Question 9

Let the I.F be xmyn.

Multiplying Differential Equation by I.F it becomes exact

⇒

This Differential Equation is Exact.

It is exact

⇒

3m - n = - 22
m + 3n = - 4

m + n = –7 + 1  = –6

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Differential Equation NAT Level - 2 - Question 10

If the P.I. of Differential Equation (x2D2 + 3xD + 1) y = 1/(1 - x)2 is of form  then, find value of λ.

Detailed Solution for Differential Equation NAT Level - 2 - Question 10

Now, let us put.

Solving above Differential Equation, we get,

Put ez = t
ezdz = dt

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