Chapter 10

Determine the order and degree (if exists) of the following differential equations:
$\frac{d y}{d x} = x + y + 5$
1. 1,1
2. -1,1
3. 1,2
4. 2,1
Express of the following physical statements in the form of differential equation.
Radium decays at a rate proportional to the amount Q present.
1. dA/dV=kQ
2. dQ/dt=kQ
3. dt/dQ=kQ
4. dV/dt=kQ
Express the following physical statements in the form of differential equation.
The population P of a city increases at a rate proportional to the product of population
and to the difference between 5,00,000 and the population.
1. dP/dt=kP(500000-P)
2. dP/dt=kP(50000-t)
3. dP/dt=kP(500000-t)
4. dP/dt=kP(50000-P)
A saving amount pays 8% interest per year, compounded continuously. In addition, the
income from another investment is credited to the amount continuously at the rate of 400 rupee per year.
1. dx/xt=2x/25+40
2. dx/xt=2x/36+400
3. dx/xt=2x/25+400
4. dx/xt=2x/36+40
Assume that a spherical rain drop evaporates at a rate proportional to its surface area. Form
a differential equation involving the rate of change of the radius of the rain drop.
1. dr/dt=k
2. dr/dt=-k
3. dr/dt=-v
4. dr/dt=v
Find the differential equation of the family of all non-vertical lines in a plane
1. $\frac{d^{2} y}{d x^{2}}$ =2
2. $\frac{d^{2} y}{d x^{2}}$ =0
3. $\frac{d^{2} y}{d x^{2}}$ =1
4. $\frac{d^{2} y}{d x^{2}}$ =5
Find the differential equation of the family of all non-horizontal lines in a plane
1. $\frac{d^{2} x}{d y^{2}}$ =5
2. $\frac{d^{2} x}{d y^{2}}$ =0
3. $\frac{d^{2} x}{d y^{2}}$ =2
4. $\frac{d^{2} x}{d y^{2}}$ =4
Find the differential equation corresponding to the family of curves represented by the
equation $y = A e^{8 x} + B e^{- 8 x}$ , where A and B are arbitrary constants.
1. $\frac{d^{2} y}{d x^{2}}$ =36y
2. $\frac{d^{2} y}{d x^{2}}$ =86
3. $\frac{d^{2} y}{d x^{2}}$ =64y
4. $\frac{d^{2} y}{d x^{2}}$ =0
Find value of m so that the function y = $e^{m x}$ is a solution of the given differential equation.
y '+ 2y = 0
1. m=-2
2. m=2
3. m=3
4. m=4
The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given that the number triples in 5 hours, find how many bacteria will be present after 10 hours?
1. After 10 hours the number of bacteria as 9 times the original number of bacteria
2. After 10 hours the number of bacteria as 5 times the original number of bacteria
3. After 10 hours the number of bacteria as 8 times the original number of bacteria
4. After 10 hours the number of bacteria as 10 times the original number of bacteria
Water at temperature 100C cools in 10 minutes to 80C in a room temperature of 25C .
Find: The temperature of water after 20 minutes
${\log_{e} \frac{11}{15} = - 0.3101; \log_{e} 5 = 1.6094}$
1. 65.33C
2. 65.03C
3. 60.33C
4. 65.43C
Water at temperature 100C cools in 10 minutes to 80C in a room temperature of 25C .
Find:
The time when the temperature is 40C
${\log_{e} \frac{11}{15} = - 0.3101; \log_{e} 5 = 1.6094}$
1. 53.46mts
2. 53.06mts
3. 43.46mts
4. 53.406mts
Suppose a person deposits 10,000 Indian rupees in a bank account at the rate of 5% per annum compounded continuously. How much money will be in his bank account 18 months later?
1. P=1000 $e^{0.075}$
2. P=10000 $e^{0.075}$
3. P=10 $e^{0.075}$
4. P=100000 $e^{0.075}$
At 10.00 A.M. a woman took a cup of hot instant coffee from her microwave oven and
placed it on a nearby Kitchen counter to cool. At this instant the temperature of the coffee
was 180 F, and 10 minutes later it was 160 F . Assume that constant temperature of the
kitchen was 70 F . What was the temperature of the coffee at 10.15A.M.?
1. 151F
2. 152F
3. 151C
4. 152C
A pot of boiling water at 100C is removed from a stove at time t = 0 and left to cool in the
kitchen. After 5 minutes, the water temperature has decreased to 80C , and another 5 minutes
later it has dropped to 65C . Determine the temperature of the kitchen.
1. 11
2. 12
3. 13
4. 14
The order of the differential equation of all circles with centre at (h, k ) and radius ‘a’ is
1. 2
2. 3
3. 4
4. 1
The number of arbitrary constants in the particular solution of a differential equation of third order is
1. 3
2. 2
3. 1
4. 0
The number of arbitrary constants in the general solutions of order n and n +1are respectively
1. n −1, n
2. n, n +1
3. n +1, n + 2
4. n +1, n
If the solution of the differential equation $\frac{d y}{d x} = \frac{a x + 3}{2 y + f}$ represents a circle, then the value of a is
1. 2
2. -2
3. 1
4. -1
P is the amount of certain substance left in after time t. If the rate of evaporation of the
substance is proportional to the amount remaining, then
1. P=C $e^{k t}$
2. P=C $e^{- k t}$
3. P=Ckt
4. Pt=C

Chapter 10

Quiz