bioblade said:Homework Statement
Solve: 3dy/dx+y=(1-2x)y^4
Homework Equations
None, really.
The Attempt at a Solution
y'+y/3=((1-2x)/3)y^4
y'/y^4+1/3y^3=(1-2x)/3
w=1/y^3
w'=(-3/y^4)y'
w'/-3+w/3=(1-2x)/3
w'-w=2x-1
e^int(-dx)=e^-x
e^(-x)*w=-e^-x(2x-1)+c
w=1-2x+c*e^x
y^3=1/(1-2x+c*e^x)
wolfram alpha says it should be -1/(-c*e^x+2x+1) though. What did I do wrong?
Bernoulli Differential Equation is a type of first-order nonlinear ordinary differential equation that involves both derivatives and powers of the dependent variable. It can be written in the form of y' + p(x)y = q(x)y^n, where n is a constant and p(x) and q(x) are functions of x.
Bernoulli Differential Equation is important in mathematics and physics as it can be used to model a wide range of real-world phenomena, such as population growth, chemical reactions, and fluid dynamics. It also has applications in engineering and economics.
The general solution to Bernoulli Differential Equation involves using a substitution to transform it into a linear differential equation, which can then be solved using standard techniques such as separation of variables or integrating factors. The particular solution can be found by using initial conditions or boundary conditions.
Yes, Bernoulli Differential Equation can be solved using numerical methods such as Euler's method, Runge-Kutta methods, or the shooting method. These methods are useful in cases where an analytical solution is not feasible.
Bernoulli Differential Equation can be used to model the spread of infectious diseases, the growth of a population, the flow of blood in arteries, and the motion of a projectile in air resistance. It can also be applied to chemical reactions, economics, and engineering problems involving non-Newtonian fluids.