How do I find the general solution to a differential equation ?

Charlotte Bryar: There are no general methods to solve any differential equation, it's a case-by-case problem. Of course, many cases can be of the same kind, for which specific methods can be used to solve them, which is how you learn about them academically.A simple example would be an equation of the kind: y ' = k, were k is a constant. So, which functions are so that they're derivative is constant? Here, simply taking the primitive is enough: y = kx + constant.In the case of the example you gave:If 2t y ' + 4t = 3, where it is understood that the differentiation is relative to t, then note that:2t y ' = 3 - 4y, or:4 y ' / (3 - 4y) = 2/tHere, someone used to calculate primitives would see the log on the left-hand side (as well as on the right). Noting that [ln|3-4y|] ' = -4 y ' / (3 - 4y), we then have:- [ln|3-4y|] ' = 2/t = 2 [ln |t|] ', or:[ln{ |3-4y| t^2 }] ' = 0ln{ |3-4y| t^2 } = k a constant|3-4y| t^2 = c a positive constant, and this comes down to the solution you! were given above.(Note that in the above, one could rewrite right away y ' / (3 - 4y) = 1/(2t) as: 4 dy / (3 - 4y) = 2 dt / t )Solving differential equations is a vast subject, and sometimes we can't find a solution (by ignorance or enormous complexity), which may lead us to numerical methods requiring monstrous computer resources (e.g. meteorological models)....Show more