Ordinary Differential Equations

The rate at which the ice melts is proportional to the amount of ice at the instant. Find the amount of ice left after 2 hours if half the quantity melts in 30 minutes. Solution. Let m be the amount of ice at any time t.

Expert Solution

∴dmdt=km⇒dmm=kdt∫dmm=k∫dt+C⇒log⁡m=kt+C At t=0,m=M:log⁡M=0+C⇒C=log⁡M On putting the value of C, (1) becomes, log⁡m=kt+log⁡Mm=M2 when t=12 hour log⁡M2=k2+log⁡M⟹log⁡M2M=k2⟹log⁡12=k2 or k=2log⁡12 On putting the value of k in (2), we have log⁡m=(2log⁡12)t+log⁡M On putting t=2 hours in (3), we have log⁡m=4log⁡12+log⁡M⇒log⁡mM=log⁡(12)4 or mM=116 or m=M16 After 2 hours, amount of ice left \( \frac{m}{M}=\frac{1}{16} \)of the amount of ice at the beginning.

Therefore. \( \frac{m}{M}=16 \)

SUN BUNRA answered 2 years ago

Ordinary Differential Equations

Solve the following problems. \( \frac{d y}{d t} \tan y=\sin (t+y)+\sin (t+y) \)

Ordinary Differential Equations

Solve the following problems. (a) \( \frac{d y}{d t}=2+\frac{y}{t} \) (b) \( 3 t+2 y \frac{d y}{d t}=y \)

Ordinary Differential Equations

Solve the following problems. \( t y \frac{d y}{d t}=\sqrt{t^{2} y^{2}+t^{2}+y^{2}+1} \)

Ordinary Differential Equations

Write a differential equation in wich \( y^2=4(t+1) \) is a solutiion.

Ordinary Differential Equations

Solve the following problems. \( \cos (2 t+y) d y=d t \)

Ordinary Differential Equations

Solve the following problems. \( (2 t-y-1) d y=(3 t+y-4) d t \)

Ordinary Differential Equations

The rate at which a body cools is proportional to the difference between the temperature of the body and that of the surrounding air. If a body in air at 25°C will cool from 100° to 75° in one minute, find its temperature at the end of three minutes.

Ordinary Differential Equations

If the population of a country doubles in 50 years, in how many years will it treble, assuming that the rate of increase is proportional to the number of inhabitants?

Ordinary Differential Equations

The number of bacteria in a certain culture grows at a rate that is proportional to the number present. If the number increased from 500 to 2000 in 2 hours, determine. (a) The number present after 12 hours. (b) The doubling time.

Ordinary Differential Equations

Suppose a student carrying a flu virus returns to an isolated college campus of 1000 students. If it is assumed that the rate at which the virus spreads is proportional not only to the number x of infected students but also to the number of students not infected, and assume that no one leaves the campus throughout the duration of the disease, determine the number of infected students after 6 days if it is further observed that after 4 days...

Question

Ordinary Differential Equations

Solutions

Expert Solution

Related Solutions

Ordinary Differential Equations

Ordinary Differential Equations

Ordinary Differential Equations

Ordinary Differential Equations

Ordinary Differential Equations

Ordinary Differential Equations

Ordinary Differential Equations

Ordinary Differential Equations

Ordinary Differential Equations

Ordinary Differential Equations