Your work might not be right, but I might be misreading the problem. Does y((3t^2)-1) represent the product of y and 3t2 - 1? It could also be interpreted as a composite function rather than a product.maximade said:Homework Statement
dy/dt=y((3t^2)-1), y(1)=-2
Homework Equations
Basic integrals
The Attempt at a Solution
integrate on both sides: dy/y=dt((3t^2)-1)
Assuming for the moment that the right side of your equation is y * (3t2 - 1), the line above should bemaximade said:========>ln(y)=(t^3)-t+c
The two lines above should bemaximade said:========>y=e^((t^3)-t+c)
========>y=e^((t^3)-t)e^(c)
maximade said:I am not sure if its some e rule that I forgot but how do you get -2 from y(1), since e^C is always positive?
A first order linear differential equation is a mathematical equation that relates a function and its derivative. It can be written in the form y' + p(x)y = q(x), where p(x) and q(x) are functions of x.
To solve a first order linear differential equation, you can use the method of integrating factors. This involves multiplying both sides of the equation by a suitable integrating factor, which is typically e∫p(x)dx. This will reduce the equation to the form (e∫p(x)dxy)' = q(x)e∫p(x)dx, which can then be integrated to find the solution for y.
The general solution to a first order linear differential equation is a family of functions that satisfy the equation. It can be found by integrating both sides of the equation and including an arbitrary constant C in the solution. The value of C can be determined by applying initial conditions or boundary conditions to the equation.
First order linear differential equations are commonly used in physics, engineering, and other fields to model real-world problems. They can be used to describe the rate of change of a physical quantity over time, such as the growth of a population, the decay of a radioactive substance, or the flow of a fluid through a pipe.
Some common applications of first order linear differential equations include modeling population growth, chemical reactions, electrical circuits, and heat transfer. They are also used in economics to model supply and demand, in biology to model growth and decay of biological systems, and in finance to model interest and investment rates.