Differential Equations Answers

Find the orthogonal trajectory curve of all the lines passing through the origin that is y = C

A tightly stretched string of length"l"has it's ends fastened at x=0andx=l. The midpoint of the string is taken the height h and released. find the initial displacement function at timet=0?

Solve by the power series method d²y/dx² -y =0

dx/dt=-4x+y+z

dy/dt=x+5y-z

dz/dt=y-3z

1. A certain substance was placed inside a room where the temperature is 17°C. it is observed that after 30 seconds, the temperature of the substance drops to 27°C and after 1 minute, the temperature drops to 20°C. what is the initial temperature of the body? Ans. 50.33°C
2. The bureau of census record in 1975 shows that the population in the country doubles compared to that of 1955. In what year will the population quadruple? Ans. 1995

1. Initially, there are 30 grams of Astatine, a highly radioactive element made in nuclear reactors. The half life of the element is 8 hours. Find the amount of substance remains for the first 5 hours. How long will it take for the 99.9% of its mass to decay? Ans. 19.45g; 79.73 hours
2.  The population of a community will be treble by the year 2009. if the population in the year 1998 initially has 2025, find the year when the population will be doubled. What is the population in the year 2012?  Ans. 2005; 8198

1. The bureau of census record in 1975 shows that the population in the country doubles compared to that of 1955. In what year will the population quadruple? Ans. 1995
2. The man puts a total amount of Php 30,000.00 to an account. For the first three years, the bank pays 4% compounded continuously and for the succeeding 7 years, the bank pays 5.5%, compounded continuously. How much will he have for the first 10 years? Ans. Php 49,709.57

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.

Solve( D^2+ DD'-6D'^2)z= cos(2x+y)

(4+x^2)dy +4dx=0