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How to solve homogeneous second order differential equations

MIT 6.002 Circuits and Electronics, Spring 2007
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Use Laplace transforms..quickest way to solve them according to me!!!
you have to asign the equation a characteristic polynomial: for example: 2 d2y/dx2 + 4 dy/dx + 6 = 0 2 m2 + 4m +6 =0 and then you solve the equation for m, you'll have two solutions since it is a second order polynomial: -if m1 = m2 the solution, will have the form Ae^m1*x + Bx e^m1*x -if m1 is different to m2 and both solutions are real, the solution will have the form A e^m1*x + B e^m2*x -if m1 is different to m2 and they both are complex numbers, you use the same form as if they were real numbers, but then you'll have to apply the euler's formula

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