There is really no need to write both c1 and c2. It's perfectly proper to immediately combine the two "constants of integration" to write ln|m|= s+ k.nick.martinez said:dm/ds=m ; m(1)=7
when i find the diff eq
∫dm/m=∫ds
ln|m|+c1=s+c2 ; k=c2-c1
A separable differential equation is an equation that can be written in the form dy/dx = f(x)g(y), where f(x) and g(y) are functions of x and y, respectively. This means that the dependent variable and independent variable can be separated on opposite sides of the equation.
To solve a separable differential equation, you must start by separating the variables on opposite sides of the equation. Then, integrate both sides with respect to their respective variables. This will result in a general solution, which can be further simplified by using any initial conditions given in the problem.
Separable differential equations are commonly used in physics, engineering, and other scientific fields to model various physical phenomena. They can be used to describe the rate of change of a quantity over time, such as the growth of a population or the decay of a radioactive substance.
No, not all differential equations can be solved using separation of variables. This method only works for equations that are in the form dy/dx = f(x)g(y). Other methods, such as substitution or variation of parameters, may be necessary to solve other types of differential equations.
One limitation of using separation of variables is that it can only be applied to first-order differential equations. Additionally, it may not always be possible to find a closed-form solution using this method, especially for more complex equations. In these cases, numerical methods may be used to approximate the solution.