Differential Equations Answers

the solution of the given linear differential equation x^2 y'+x(x+2)=e^x for x>0

1. A radioactive substance plutonium 239 has a half life of 24100 years. Initially, there is 30mg of the substance, find how much will remain for the first 1000 years. How long for the substance to decay 90% of its initial mass? Ans. 29.15mg; 80,058.47 years
2.  A thermometer reading of 50°C was plunged into a tub of frozen water. If the thermometer reads 30° after 5 seconds, what will be the reading after 10 seconds? How long will it take for the thermometer reading to drop to 20°C? Ans. 18°C; 8.97 seconds

Uxx + Uyy =0 convert the situation equation into its Canonical form and find out its general solution

1. In 8:00 AM, the population of a bacteria is 1000. At 10:30 AM, the number of bacteria triples. At what time will the population become 100 times the initial population of bacteria? What will be the population at 2:00 PM?  Ans. 6:29 PM; 13,967
2.  If the present population of a certain country is 40 million and in 10 years, the population is 50 million, what will be its population 20 years from now? Ans. 62.5 million

Uxx + Uyy =0 convert the situation equation into its Canonical form and find out its general solution

Determine the general solution to the equation ∂²u/∂t²=c²(∂²u/∂x²) under the boundary conditions u(0,t)=u(1,t)=0 and initial conditions u(x,0)=Φ(x), u_t(x,o)=Ψ(x)

Obtain the differential equation that describe the family of curve.

1. All straight lines tangent to a unit circle with center at (1, 1)

dy/dx=xe^x/(e^y+x^2e^y)

under the same assumptions underlying the model in (1), determine a differential equation for the population P(t) of a country when individuals are allowed to immigrate into the country at a constant rate r>0. What is the differential equation for the population P(t) of the country when individuals are allowed to emigrate from the country at a constant rate r>0?

Determine the general solution to the equation ∂²u/∂t²=c²(∂²u/∂x²) under the boundary conditions u(0,t)=u(1,t)=0 and initial conditions u(x,0)=Φ(x), u_t(x,o)=Ψ(x)