Rearrange so each variable is on one side and integrate both sides.
If dy/dx=g(x)h(y), rearrange to dy/h(y)=g(x)dx and integrate both sides. Remember the constant and watch for lost solutions where h(y)=0. Many physical growth and decay laws are separable.
dy/dx=ky gives ln y=kx+C, so y=Ce^{kx}.
Separable equations are the easiest ODEs to solve exactly, and they appear in many decay, cooling and population problems.
Rearrange so each variable is on one side and integrate both sides.
Separable equations are the easiest ODEs to solve exactly, and they appear in many decay, cooling and population problems.
If you can write the right-hand side as a product of a function of time and a function of the dependent variable, you can separate the variables.
Include the constant of integration once and apply the initial condition afterward. Check that you did not divide by a variable that can be zero.
Typical applications include solving Newton's law of cooling for the temperature of a cup of coffee, finding the velocity of a body with quadratic drag and solving the discharge of a tank through an orifice.
Study "First-order ODEs" first (topic 59 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Integrate each side separately. Solve for y if possible. Apply initial condition.
dy/dx=ky gives ln y=kx+C, so y=Ce^{kx}.
The next topic is "Integrating factor method", which builds directly on this one.
It is classed as Intermediate in Engineering Mathematics (topic 60 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.
Want to practise? Open the interactive course for quick checks, quizzes, simulations and progress tracking.