Differential Equation

Study Material (PDF)
Self Assesment
Study Material (PDF)
Self Assesment

Click on each question to see answer & solution.

Question No: #171

If the rate of increment of radius of a circle is `\frac{1}{\pi}`, then find the rate of change of its area when radius is 2 unit.

Question No: #172

A particle starts from the origin with a velocity 5 cm/s and moves in a straight line, its acceleration at time `t` seconds being `(3t^2-5t)` cm/`s^2`. Find velocity of the particle and its distance from the origin at the end of 4 seconds.
Ans: 29 cm/sec, 30.67 cm (approx)
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Question No: #173

If population growth rate is `5%`, in how many years, it becomes double?

Question No: #174

A radio-active element satisfies the equation `\frac{dv}{dt}=-kv` for its natural decay, where `v` is the volume of the element at time `t` and `k` is positive constant. If 40% of the substance disappear in 25 years, find the time it takes to disappear 60% of the substance.
Ans: 44.8 years (approx)
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Question No: #175

At any instant, the rate of disintegration of Uranium is proportional to its mass. If the amount of Uranium at time `t_1` and `t_2` is `m_1` and `m_2` respectively, then show that half life of uranium is `\frac{(t_2-t_1)log2}{logm_1-logm_2}`

Question No: #176

Let the rate of growth of a colony of bacteria be proportional to the square* root of the number of bacteria present at any time. When an experiment started, it was estimated that there were about 16 times of bacteria after 5 hours, then when it is 49 times? (*alt: cube)

Question No: #177

Let the rate of growth of population in a city be proportional to number of citizen at any time. When an experiment started, it was estimated that there were about double of population after 30 years, then when it is three times? Given `\log_e3=1.6\log_e2`

Question No: #178

Let `r` be the radius of any sphere. If `s` & `v` be the area of outer surface and volume of this sphere respectively, then show that `2\frac{dv}{dt}=r\frac{ds}{dt}`

Question No: #179

Spherical raindrops are evaporated. The rate of decrement of volume of raindrop is proportional to its surface area. Prove that the radius of spherical raindrop changes uniformly.

Question No: #180

The volume of a spherical balloon increases at a rate of 10 cc per second. Find the rate of change of its outer surface when its radius is 16 cm.