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Test: First Order Differential Equations (non-linear) - Civil Engineering (CE) MCQ


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10 Questions MCQ Test Engineering Mathematics - Test: First Order Differential Equations (non-linear)

Test: First Order Differential Equations (non-linear) for Civil Engineering (CE) 2024 is part of Engineering Mathematics preparation. The Test: First Order Differential Equations (non-linear) questions and answers have been prepared according to the Civil Engineering (CE) exam syllabus.The Test: First Order Differential Equations (non-linear) MCQs are made for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: First Order Differential Equations (non-linear) below.
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Test: First Order Differential Equations (non-linear) - Question 1

Solve the differential equation

Detailed Solution for Test: First Order Differential Equations (non-linear) - Question 1



Test: First Order Differential Equations (non-linear) - Question 2

Solve the differential equation

Detailed Solution for Test: First Order Differential Equations (non-linear) - Question 2


Dividing both sides by z (logz)2

Integrating Factor

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Test: First Order Differential Equations (non-linear) - Question 3

The singular solution ofis/are

Detailed Solution for Test: First Order Differential Equations (non-linear) - Question 3

The given differential equation is

xD2 – 2y D + 4x = 0

The discriminant of this equation is

(-2y)2 – 4x.4x = 0 ⇒ y2 – 4x= 0

Differentiating with respect to x,

⇒ yD = 4x ⇒ D = 4x/y

Substituting D = 4x/y

⇒ y2 – 4x2 = 0

Hence y2 = 4x2 satisfies the given differential equation. The singular solution is

y2 – 4x2 = 0 ⇒ y = ±2x

Test: First Order Differential Equations (non-linear) - Question 4

The solution of the differential equation forsatisfying the condition y(0) = 1/√3

Detailed Solution for Test: First Order Differential Equations (non-linear) - Question 4



Test: First Order Differential Equations (non-linear) - Question 5

Solve

Detailed Solution for Test: First Order Differential Equations (non-linear) - Question 5

dy/dx = p
p2 = tan⁡(y − xp)
tan−1p2 = y − xp
y = xp + tan−1p2

y = px + f(p)

Which is clairaut’s equation.

So equation is:
y = cx + tan−1c2

Test: First Order Differential Equations (non-linear) - Question 6

Consider the differential equationwith y(0) = 1. Then the value of y(1) is

Detailed Solution for Test: First Order Differential Equations (non-linear) - Question 6


Integrating factor, IF = e∫dx = ex

y(IF) = ∫(IF)exdx + C

At x = 0, y = 1
∴ C = 1/2

Test: First Order Differential Equations (non-linear) - Question 7

Solve the differential equation

Detailed Solution for Test: First Order Differential Equations (non-linear) - Question 7


Test: First Order Differential Equations (non-linear) - Question 8

The differential equation representing the family of circles touching y-axis at origin is:

Detailed Solution for Test: First Order Differential Equations (non-linear) - Question 8

Equation of circle having centre at (r, 0) and radius r is written as:

(x - r)2 + (y - 0)2 = r2

x2 + r2 - 2xr + y2 = r2

x2 - 2xr + y2 = 0

x2 + y2 = 2xr        ___(1)

Differentiating on both sides:

x + yy’ = r

Put value of r in equation (1):

Thus, x2 + y2 = 2 (x + yy’) x

2xyy’ + x2 - y2 = 0

Note:

A differential equation is said to be linear if the dependent variable and its derivative appears in first degree.

Note:

 is the first order linear differential equation.

Test: First Order Differential Equations (non-linear) - Question 9

Which of the following is one of the criterions for linearity of an equation?

Detailed Solution for Test: First Order Differential Equations (non-linear) - Question 9

The two criterions for linearity of an equation are:

  • The dependent variable y and its derivatives are of first degree.
  • Each coefficient depends only on the independent variable
Test: First Order Differential Equations (non-linear) - Question 10

Which of the following is the property of error function?

Detailed Solution for Test: First Order Differential Equations (non-linear) - Question 10

Error Function is given by, 
Some of its properties are:
erf (0) = 0
erf (∞) = 1
erf (-x) = -erf(x)

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