Question

Find the general solution of the differential equation e^x tan y dx + (1 - e^x) sec² y dy = 0.

An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form dy/dx = F(x, y) is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogenous function of degree n if F(λx, λy) = λⁿ F(x, y). To solve a homogeneous differential equation of the type dy/dx = F(x, y) = g(y/x), we make the substitution y = vx and then separate the variables.

Find the general solution of the following differential equation : dy/dx = e^(x-y) + x²e^(-y)

A bacteria sample of certain number of bacteria is observed to grow exponentially in a given amount of time. Using exponential growth model, the rate of growth of this sample of bacteria is calculated. The differential equation representing the growth of bacteria is given as : dP/dt = kP, where P is the population of bacteria at any time 't'. Based on the above information, answer the following questions :