Differential Equations Test

Result

Differential Equations Test - 2

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
1 / -0.25

The differential equation,is a:

A
second order, third degree equation.

B
second order, second degree equation.

C
third order, third degree equation.

D
second order, first degree equation

Solution

To calculate the degree or the order of a differential equation, the powers of derivatives should be an integer.
On squaring both sides, we get a differential equation with the integral power of derivatives.
⇒ Order (the highest derivative) = 2
⇒Degree (the power of highest degree) = 1
Question 2
1 / -0.25

The differential equation for the equation is :

A

B

C

D

Solution

y = Acos(αx) + Bsin(αx)
dy/dx = -A αsin(αx) + B αcos(αx)
d² y/dx² = -A α² cos(αx) - B α² sin(αx)
= -α² (Acos(αx) + Bsin(αx))
= -α² * y
d² y/dx² + α² *y = 0
Question 3
1 / -0.25

Formation of the differential equation of the family of curves represented by y = Ae^2x + Be^-2x is :

A

B

C

D

Solution

y = Ae^2x + Be^-2x
dy/dx = 2Ae^2x –2Be^-2x
d² y/dx² = 4Ae^2x + 4Be^-2x
= 4*y
d² y/dx² –4y = 0
Question 4
1 / -0.25

The degree of the differential equation

A
2

B
1

C
3

D
0

Solution

Given equation is :
(dy/dx)² + 1/(dy/dx) = 1
((dy/dx)³ + 1)/(dy/dx) = 1
(dy/dx)³ +1 = dy/dx
So, final equation is
(dy/dx)³ - dy/dx + 1 = 0
So, degree = 3
Question 5
1 / -0.25

Differential equation representing the family of curves given by y = ax + x² is:

A

B

C

D

Solution

The answer is C. We eliminate constants.
We have
y=ax+x²
Differentiating with respect to x,
Question 6
1 / -0.25

The order of the differential equation:

A
4

B
2

C
3

D
1

Solution

Order of the D.E. is 3
Order of a differential equation is the order of the highest derivative present in the equation.
Question 7
1 / -0.25

Formation of the differential equation corresponding to the ellipse major axis 2a and minor axis 2b is:

A

B

C

D

Solution

Equation of ellipse :
x² /a² + y² /b² = 1
Differentiation by x,
2x/a² + (dy/dx)*(2y/b² ) = 0
dy/dx = -(b² /a² )(x/y)
-(b² /a^2) = (dy/dx)*(y/x) ----- eqn 1
Again differentiating by x,
d² y/dx² = -(b² /a² )*((y-x(dy/dx))/y² )
Substituting value of -b² /a² from eqn 1
d² y/dx² = (dy/dx)*(y/x)*((y-x(dy/dx))/y² )
d² y/dx² = (dy/dx)*((y-x*(dy/dx))/xy)
(xy)*(d² y/dx² ) = y*(dy/dx) - x*(dy/dx)²
(xy)*(d² y/dx² ) + x*(dy/dx)² - y*(dy/dx) = 0
Question 8
1 / -0.25

The differential equation is a solution of the equation:

A
y = (A/x) + B

B
xy = (A/x) + B

C
x² y = Ax + B

D
xy = Ax –B

Solution

Solving second order differential equation with variable coefficients becomes a bit lengthy and complicated. So, its better to check by options.
On checking option A :
y = A/x + B
dy/dx = -A/x²
d² y/dx² = (2A)/x³
So,
d² y/dx² + (2/x)*(dy/dx) = 0
(2A)/x³ + (2/x)*((-A)/x² ) = 0
(2A - 2A)/x³ = 0
0 = 0
LHS = RHS
Question 9
1 / -0.25

The differential equation is a:

A
Third order, third degree equation

B
Second order, second degree equation

C
Second order, first degree equation

D
Second order, third degree equation

Solution

3*(d² y/dx² ) = [1+(dy/dx)² ]³ /²
On squaring both side,
9*(d² y/dx² )² = [1+(dy/dx)² ]³
The order of the equation is 2. The power of the term determining the order determines the degree.
So, the degree is also 2.
Question 10
1 / -0.25

The order and degree of the differential equation: (y ”)² + (y ”)³ + (y ’)⁴ + y⁵ = 0 is:

A
2, 4

B
3, 5

C
2, 5

D
2, 3

Solution

The highest order derivative here is y ’’. Therefore the order of the differential equation=2.
The highest power of the highest order derivative here is 3. Therefore the order of the differential equation=3.