JEE Advanced (Single Correct Type): Differential Equations | Chapter-wise Tests for JEE Main & Advanced PDF Download

Q.1. hat is the order of differential equation y’’ + 5y’ + 6 = 0?
(a) 0
(b) 1
(c) 2
(d) 3

Correct Answer is option (c)
Given, differential equation y’’ + 5y’ + 6 = 0. The highest order derivative present in the differential equation is y’’. Hence, the order is 2.

Q.2. The differential equation representing the family of curves y² = 2c (x + √c) , where c is a positive parameter, is of
(a) order 1
(b) order 2
(c) degree 2
(d) degree 1

Correct Answer is option (a)
We have, y2 = 2c (x + √c) ...(i)
⇒ 2y y₁ = 2c ⇒ yy₁ = c ...(ii)
Eliminating c from (i) and (ii), we get )

y² = 2yy₁(x +⇒ (y - xy₁)² = y y³₁
Clearly, it is a differential equation of order one and degree 3.

Hence, (A) is the correct answer.

Q.3. What is the degree of differential equation (y’’’)² + (y’’)³ + (y’)⁴ + y⁵ = 0?
(a) 2
(b) 3
(c) 4
(d) 5

Correct Answer is option (a)
The degree is the power raised to the highest order derivative. Therefore, in the given differential equation, (y’’’)² + (y’’)³ + (y’)⁴ + y⁵ = 0, the degree will be power raised to y’’’.
So, the answer is 2.

Q.4.  For any differentiable function y = f(x), the value of is
(a) always zero
(b) always non-zero
(c) equal to 2y²
(d) equal to x²

Correct Answer is option (a)
for a differential coefficient

Hence, (a) is the correct answer.

Q.5. Find the order of differential equations:
(a) 2
(b) 1
(c) 0
(d) Undefined

Correct Answer is option (a)
Given, the differential equation is:

Or we can write:
2x² y’’ – 3y’ + y = 0
Order is the highest derivative in the differential equation. Therefore, the order is 2.

Q.6. Find the degree of the differential equation.

(a) 0
(b) 1
(c) 2
(d) 3

Correct Answer is option (d)
Given, the differential equation is:

We can expand it and get:

The exponent of highest derivative is the degree. Therefore, the degree is 3.

Q.7.  The number of arbitrary constants in the particular solution of a differential equation of third order is:
(a) 3
(b) 2
(c) 1
(d) 0

Correct Answer is option (d)
The solution free from arbitrary constants i.e., the solution obtained from the general solution by giving particular values to the arbitrary constants is called a particular solution of the differential equation.

Q.8. Which of the following is a second-order differential equation?
(a) (y’)² + x = y²
(b) y’y” + y = sin x
(c) y”’ + (y”)² + y = 0
(d) y’ = y²

Correct Answer is option (b)
The order of y’y” + y = sin x is 2. Thus, it is a second-order differential equation.

Q.9. The function f(θ) = satisfies the differential equation
(a)
(b)
(c)
(d)  

Correct Answer is option (b)
We have f(θ) =
Therefore df(θ) = -2cosec²θ cotθ
Hence, (a) is the correct answer.

Q.10. If f(x), g(x) be twice differentiable function on [0, 2] satisfying f "(x) = g ''(x), f '(1) = 4 and g'(1) = 6, f(2) = 3, g(2) = 9, then f(x) - g(x) at x = 4 equals
(a) 0
(b) -10
(c) 8
(d) 2

Correct Answer is option (b)
We have f ''(x) = g ''(x)
Integrating, we get f '(x) = g '(x)+ c
Putting x = 1, we get f '(1) = g '(1) + c ⇒ c = -2
⇒ f'(x) = g'(x) - 2  ⇒ f(x) = g(x) - 2x + C₁  ⇒ f(2) = g(2) - 4 + C₁ ⇒ C₁ = -2
Thus we have f(x) = g(x) - 2x - 2
⇒ f(4) - g(4) = -10
Hence, (B) is the correct answer.

Q.11. The degree and order of the differential equation of all parabolas whose axis is x-axis are
(a) 2, 1
(b) 1, 2
(c) 3, 2
(d) none of these

Correct Answer is option (b)
Equation of required parabola is of the form y² = 4a(x –h)
Differentiating, we have
Required differential equation
Degree of the equation is 1 and order is 2.
Hence, (B) is the correct answer.

Q.12. Solution of

(a) sin (y/x) = kx
(b) cos y/x = kx
(c) tan y/x = kx
(d) none of these

Correct Answer is option (b)

put y = vx ⇒ v + = v + tan v 

cot v dv = dx/v

Integrating, we get ln sin v = ln x + ln k ⇒ sin y/x = kx
Hence, (A) is the correct answer.

Q.13. If y = e^–x (A cosx + B sin x), then y satisfies
(a)
(b)
(c)
(d)

Correct Answer is option (c)
y = e^–x (A cos x + B sin x) ....(1)
⇒ dy/dx =  e^-x (–A sin x + B cos x) – e^–x (A cos x + B sin x)
⇒ dy/dx =  e^-x (–A sin x + B cos x) – y .....(2)
⇒ d²y/dx² = e^-x  (–A cos x – B sin x) – e^–x (–A sin x + B cos x)

Using (1) and (2), we get

Hence, (C) is the correct answer.

Q.14. If  = y (logy – logx + 1) then the solution of the equation is:
(a)
(b)
(c)
(d)

Correct Answer is option (d)

Put y = vx  ⇒

⇒ log(log v) = logx + logc = logcx
⇒ log v = cx  ⇒= cx
Hence, (D) is the correct answer.

Q.15. The differential equation of all ellipses centerd at the origin and major and minor axes along coordinate axes is
(a) y² + xy₁² - yy₁ = 0
(b) xyy₂ + xy₁² -yy₁ = 0
(c) yy₂ + xy₁² - xy₁ = 0
(d) none of these

Correct Answer is option (b)
The given family of curves is
...(1)
where a and b are parameters
...(2)
Differentiating again, we have
 ..(iii)
Multiplying (iii) with x and then subtracting from (ii), we have
(1/b²) (yy₁ - xy₁² - xyy₂) = 0 ⇒ xyy₂ + xy₁² - yy₁ = 0