I have solved Linear first order differential equations, I know how to separate variables, for higher orders I know the the substitutions that apply to the bernoulli equation and homogeneous equations.

The exercise asks to solve:

$$y^{(4)}+ 4y^{(2)}+4y = cos2t - sin2t$$

I am given a hint that : $$r^4+4r^2 + 4r^0 = 0 \implies (r^2+2)^2 = 0 \implies r = \pm i \sqrt{2}$$

Could I have a reference to how to solve these kind of differential equations? Or am I just missing a trick that makes this a first order? I tried thinking about substituting Eulers formula in the left hand side but I am missing an $i$.

  • 1
    $\begingroup$ I believe this is what you're looking for. $\endgroup$
    – Git Gud
    Commented Oct 18, 2014 at 15:46

1 Answer 1


hint: use the fact that $e^{i\sqrt{2}t}=\cos(\sqrt{2}t)+i\sin(\sqrt{2}t)$


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