Express each of the following physical statements in the form of differential equation.
 

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.

Show that y = e^−x + mx + n is a solution of the differential equation e^x \(\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \) -1 = 0

Show that y = ae^-3x + b, where a and b are arbitary constants, is a solution of the differential equation\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3\frac { dy }{ dx } =0\)

Show that y = ax + \(\frac { b }{ x } \), x ≠ 0 is a solution of the differential equation x² y" + xy' - y = 0.

Show that the differential equation representing the family of curves \({ y }^{ 2 }=2a\left( x+a^\frac { 2 }{ 3 } \right) \) where a is a positive parameter, is \({ \left( { y }^{ 2 }-2xy\frac { 2 }{ 3 } \right) }^{ 3 }=8{ \left( y\frac { dy }{ dx } \right) }^{5 }\).

Show that y = a cos bx is a solution of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ b }^{ 2 }y=0\).

If F is the constant force generated by the motor of an automobile of mass M, its velocity is given by M \(\frac{dV}{dt}\)= F-kV, where k is a constant. Express V in terms of t given that V = 0 when t = 0.

The velocity v , of a parachute falling vertically satisfies the equation \(\\ \\ \\ \\ \\ \\ \\ v\frac { dv }{ dx } =g\left( 1-\frac { { v }^{ 2 } }{ { k }^{ 2 } } \right) \\ \\ \), where g and k are constants. If v and x are both initially zero, find v in terms of x.

Find the equation of the curve whose slope is \(\frac { y-1 }{ { x }^{ 2 }+x } \) and which passes through the point (1, 0).

Solve the following differential equations or show that the solution of 
\(\\ \\ \\ \frac { dy }{ dx } =\sqrt { \frac { 1-{ y }^{ 2 } }{ 1-{ x }^{ 2 } } } \)

Solve the following differential equations:
ydx + (1 +x²) tan^-1 xdy = 0

Solve the following differential equations:
\(sin\frac { dy }{ dx } =a,y(0)=1\)

Solve the following differential equations:
\(\frac { dy }{ dx } ={ e }^{ x+y }+{ x }^{ 3 }{ e }^{ y }\)

Solve the following differential equations:
(e^y+1) cos x dx + e^y sin x dy = 0

Solve the following differential equations:
(ydx-xdy)cot\(\left( \frac { x }{ y } \right) \) = ny² dx

Solve the following differential equations:
\(\frac { dy }{ dx } -x\sqrt { 25-{ x }^{ 2 } } =0\)

Solve the differential equation:
x cos y dy = e^x(x log x + 1)dx

Solve the following differential equations:
tan y\(\frac{dy}{dx}\) = cos(x+y)+cos(x-y)

Solve the following differential equations:
\(\\ \\ \\ \frac { dy }{ dx } ={ tan }^{ 2 }(x+y)\)

Solve (2x + 3y)dx + (y − x)dy = 0.

Solve the following differential equations
\(\left[ x+y\quad cos\left( \frac { y }{ x } \right) \right] dx=x\ cos\left( \frac { y }{ x } \right) dy\)

Solve the following differential equations
(x³+ y³) dy-x²ydx = 0

Solve the differential equation \({ ye }^{ \frac { x }{ y } }dx=\left( { xe }^{ \frac { x }{ y } }+y \right) dy\)

Solve the following differential equations 2xydx + (x² + 2y²)dy = 0

Solve the differential equation (y²-2xy) dx = (x²-2xy) dy

Solve the following differential equations
\(x\frac { dy }{ dx } =y-x{ cos }^{ 2 }\left( \frac { y }{ x } \right) \)

Solve the following differential equations
\(\left( 1+3{ e }^{ \frac { y }{ x } } \right) dy+3{ e }^{ \frac { y }{ x } }\left( 1-\frac { y }{ x } \right) dx=0,\) given that y = 0 when x = 1

Solve the following differential equations
(x²+y²)dy = xy dx. It is given that y(1) = 1 and y(x₀) = e. Find the value of x₀.

Solve: \(\frac{dv}{dx}+2y\ cot\ x=3x^2 cosec^2x\)

Solve (1+x³)\(\frac { dy }{ dx } \)+ 6x²y = 1+x².

Solve ye^ydx = (y³+2xe^y)dy

Solve the Linear differential equation:
\(({ x }^{ 2 }+1)\frac { d }{ y } dx+2xy=\sqrt { { x }^{ 2 }+4 } \)

Solve the Linear differential equation:
(2x- 10y³) dy + ydx = 0

Solve the Linear differential equation:
x sin x \(\frac { dy }{ dx }\) + (x cos x + sin x) y = sinx

Solve the Linear differential equation:
\(\left( y-{ e }^{ sin^{ -1 }x } \right) \frac { dx }{ dy } +\sqrt { 1-{ x }^{ 2 } } =0\)

Solve the Linear differential equation:
\(\frac { dy }{ dx } +\frac { y }{ (1-x)\sqrt { x } } =1-\sqrt { x } \)

Solve the Linear differential equation \((1+x+{ xy }^{ 2 })\frac { dy }{ dx } +(y+{ y }^{ 3 })=0\)

Solve the Linear differential equation:
\(\frac { dy }{ dx } +\frac { y }{ xlogx } =\frac { sin2x }{ logx } \)

Solve the Linear differential equation:
\((x+a)\frac { dy }{ dx } -2y={ (x+a) }^{ 4 }\)

Solve the Linear differential equation:
\(\frac { dy }{ dx } =\frac { { sin }^{ 2 }x }{ 1+{ x }^{ 3 } } -\frac { { 3x }^{ 2 } }{ 1+{ x }^{ 3 } } y\)

Solve the Linear differential equation:
\(x\frac { dy }{ dx } +y=xlogx\)

Solve the Linear differential equation:
\(x\frac { dy }{ dx } +2y-x^2logx=0\)

Solve the Linear differential equation:
\(\frac { dy }{ dx } +\frac { 3y }{ x } =\frac { 1 }{ { x }^{ 2 } } \), given that y = 2 when x = 1 

The growth of a population is proportional to the number present. If the population of a colony doubles in 50 years, in how many years will the population become triple?

A radioactive isotope has an initial mass 200mg, which two years later is 50mg. Find the expression for the amount of the isotope remaining at any time. What is its half-life? (half-life means the time taken for the radioactivity of a specified isotope to fall to half its original value).

In a murder investigation, a corpse was found by a detective at exactly 8 p.m. Being alert, the detective also measured the body temperature and found it to be 70^oF. Two hours later, the detective measured the body temperature again and found it to be 60^oF. If the room temperature is 50^oF, and assuming that the body temperature of the person before death was 98.6^oF, at what time did the murder occur? [log(2.43) = 0.88789; log(0.5)=-0.69315]

A tank contains 1000 litres of water in which 100 grams of salt is dissolved. Brine (Brine is a high-concentration solution of salt (usually sodium chloride) in water) runs in a rate of 10 litres per minute, and each litre contains 5 grams of dissolved salt. The mixture of the tank is kept uniform by stirring. Brine runs out at 10 litres per minute. Find the amount of salt at any time t.

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?

Find the population of a city at any time t, given that the rate of increase of population is proportional to the population at that instant and that in a period of 40 years the population increased from 3,00,000 to 4,00,000.

The equation of electromotive force for an electric circuit containing resistance and self inductance is E = Ri + L\(\frac{di}{dt},\) Where E is the electromotive force is given to the circuit, R the resistance and L, the coefficient of induction. Find the current i at time t when E = 0.

The engine of a motor boat moving at 10 m/s is shut off. Given that the retardation at any subsequent time (after shutting off the engine) equal to the velocity at that time. Find the velocity after 2 seconds of switching off the engine.

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?

Assume that the rate at which radioactive nuclei decay is proportional to the number of such nuclei that are present in a given sample. In a certain sample 10% of the original number of radioactive nuclei have undergone disintegration in a period of 100 years. What percentage of the original radioactive nuclei will remain after 1000 years?

Water at temperature 100^oC cools in 10 minutes to 80^oC in a room temperature of 25^oC.
Find
(i) The temperature of water after 20 minutes
(ii) The time when the temperature is 40^oC
\(\left[ { log }_{ e }\frac { 11 }{ 15 } =-0.3101;{ log }_{ e }5=1.6094 \right] \)

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^o F, and 10 minutes later it was 160^o F. Assume that constant temperature of the kitchen was 70^oF.
(i) What was the temperature of the coffee at 10.15 A.M.? \(\left[\log \frac{9}{11}=-0.6061\right]\)
(ii) The woman likes to drink coffee when its temperature is between 130^oF and 140^oF between what times should she have drunk the coffee? \(\left[\log \frac{6}{11}=-0.2006\right]\)

A pot of boiling water at 100^o 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^o C , and another 5 minutes later it has dropped to 65^oC. Determine the temperature of the kitchen.

A tank initially contains 50 litres of pure water. Starting at time t = 0 a brine containing with 2 grams of dissolved salt per litre flows into the tank at the rate of 3 litres per minute. The mixture is kept uniform by stirring and the well-stirred mixture simultaneously flows out of the tank at the same rate. Find the amount of salt present in the tank at any time t > 0.