Differential Equations Answers

Let 𝑦1 and 𝑦2 be linearly independent solutions of the differential equation 𝑦 ′′ + 𝑝(𝑥)𝑦 ′ + 𝑞(𝑥)𝑦 = 0, where functions 𝑝 and 𝑞 are continuous on some interval 𝐼. (i) Prove that 𝑊(𝑦1, 𝑦2 )(𝑥) = 𝐶𝑒 − ∫ 𝑝(𝑥)𝑑𝑥 , where 𝑊 is the Wronskian and 𝐶 ∈ ℝ is an arbitrary constant.

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)