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There are several ways to solve dependent source circuit problems. The two most common methods used while learning circuit analysis are the linear superposition method and the transfer function method.

The linear superposition method is the most straight-forward. Assuming the circuit is linear, you simply set up a system of linear equations corresponding to each dependent source, and solve the equations. There are numerous methods of solving systems of linear equations, all of which are covered in the branch of mathematics known as linear algebra.

The transfer function method, in actuality there are many of these types of methods, either turns multiple dependent source problems into a single dependent source problem or changes the domain in which you're working from the time domain into a far simpler, mathematically equivalent domain parameter. Laplace transforms are a good example of this type of method.

I've included a bunch of links if you want to learn more.

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