Differential Equations Test

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Differential Equations Test - 5

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Question 1
1 / -0.25

Degree of a differential equation, when the equation is polynomial equation in y ′is

A
Highest (positive integral index) of the lowest order derivative in the given differential equation.

B
Lowest power (positive integral index) of the highest order derivative in the given differential equation.

C
Lowest power (positive integral index) of the lowest order derivative in the given differential equation.

D
Highest power (positive integral index) of the highest order derivative in the given differential equation.

Solution

Degree of a differential equation, when the equation is polynomial equation in y ′isHighest power (positive integral index) of the highest order derivative in the given differential equation.
Question 2
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The number of arbitrary constants in the general solution of a differential equation of fourth order are:

A
3

B
4

C
1

D
2

Solution

4 , because the no. of arbitrary constants is equal to order of the differential equation.
Question 3
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For the differential equation xy = (x+2) (y+2) find the solution curve passing through the point (1, –1).

A
y + x + 2 = log (x² (y + 2)² )

B
y −− x − 2 = log (x² (y + 2)² )

C
y −− x + 2 = log (x² (y + 2)² )

D
y −− x − 2 = log (x² (y − 2)² )

Solution
Question 4
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Differential equation of the family of ellipses having foci on y-axis and centre at origin is

A
xy ′′ + x(y ′)²   yy ′ = 0

B
xyy ′′ + x(y ′)²   yy ′ = 0

C
yy ′′ + x(y ′)²   yy ′ = 0

D
xyy ′′ −x(y ′)² +yy ′ = 0

Solution
Question 5
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A
x = sin 2y+C

B
x = sin y+C

C
x = cos y+C

D
None of these

Solution
Question 6
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The order of the equation

A
1

B
4

C
3

D
2

Solution

2
Question 7
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The number of arbitrary constants in the particular solution of a differential equation of third order are:

A
2

B
0

C
3

D
1

Solution

0 , because the particular solution is free from arbitrary constants.
Question 8
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Find the particular solution of the differential equation log = 3x+4y, given that y = 0 and x = 0.

A
4e^3x +3e^−4y +7 = 0

B
4e^3x −3e^−4y −7 = 0

C
4e^3x +3e^−4y −7 = 1

D
4e^3x +3e^−4y −7 = 0

Solution
Question 9
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Find the equation of a curve passing through the point (0, 0) and whose differential equation is y ′= ex sin x.

A
4y − 1 = e^x ( sin x − cos 2x)

B
3y − 1 = e^x ( sin x − cos2x)

C
2y − 1 = e^x ( sin x −cos x)

D
2y + 1 = e^x ( sin2 x − cos x)

Solution
Question 10
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A
y = x log |3x| +Cx

B
y = log |x| +Cx

C
y = log |2x| +Cx

D
y = x log |x|+Cx

Solution
Question 11
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The order of the equation

A
3

B
2

C
4

D
1

Solution

3
Question 12
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General solution of

A
y = 2tan (x/2) − x+ c

B
y = 2tan² (x/2) − x+ c

C
y = 2tan (x) –x + c

D
y = 2tan (x/2) + x+ c

Solution
Question 13
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In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs 100 double itself in 10 years (loge2 = 0.6931).

A
7.93%

B
8.93%

C
9.93%

D
6.93%

Solution

Let P be the principal at any time t. then,

When P = 100 and t = 0., then, c = 100, therefore, we have:

Now, let t = T, when P = 100., then;
Question 14
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Find the equation of a curve passing through the point (0, –2) given that at any point (x, y) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.

A
y³ − x² = 4

B
y² − x² = 4

C
y³ − x³ = 4

D
y² − x³ = 4

Solution

When x= 0 and y = - 2 ,we have : c = 2,
Question 15
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Solution of (x –y) dy –(x + y) dx = 0 is

A

B

C

D

Solution