Solving PDE, ODE by the integral transformation

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I was solving some PDE's and ODE'S via Fourier/Laplace trasform for my class, because it is said it can be useful (easier to solve). My exercices usually contain pretty easy differential equations and I would love to see some differential equation where the difference is big (like hard to solve classicaly, easy by the transform) and I can't think about any example because this subject is relatively new for me and I am trying to build my intuition. Can anyone help me to come up with such an example?