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BarackObama
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Homework Statement
dz/dt + e^(t+z) = 0
Homework Equations
The Attempt at a Solution
dz/dt = -e^te^z
integral(dz/e^z) = integral(-e^tdt)
let u = 1/e^z
dv = dz
du = -e^-zdz v= z
integral(udv)
= z/e^z + integral(ze^-zdz)
BarackObama said:integral(dz/e^z) = integral(-e^tdt)
A separable differential equation is a type of differential equation where the variables can be separated and solved independently. This means that the equation can be rewritten in the form of dy/dx = f(x)g(y), where f(x) is a function of x and g(y) is a function of y.
To solve a separable differential equation, you first rearrange the equation so that all terms with y are on one side and all terms with x are on the other side. Then, you integrate both sides with respect to their respective variables. This will result in an implicit equation, which can be solved for y to obtain the solution.
Separable differential equations are commonly used in physics, engineering, economics, and other fields to model various physical processes. For example, they can be used to model population growth, radioactive decay, and chemical reactions.
No, not all differential equations can be solved using the separable method. This method only works for equations that can be written in the form of dy/dx = f(x)g(y). Other types of equations, such as exact equations or non-separable first-order equations, require different methods to solve.
One limitation of the separable method is that it only works for first-order differential equations. It is also not always possible to find an explicit solution using this method, and sometimes only implicit solutions can be obtained. Additionally, the separable method may not be applicable to certain special cases of differential equations, such as singular or degenerate equations.