Laplace Transform: Explaining Theory & Solving ODE

marioooo
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Hello,

What does laplace transformation exactly 'do'? If I have PDE of second order and use LT on it, what do i get to solve? ODE? or if I have ODE of second order, what do I need to solve afet transformation? How does this work? is there any rule?!
 
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The thing with integral transforms is that they can turn differential equations into algebraic equations which are far easier to solve and then you can transform back to obtain the solutions of your differential equations.
 
Thread 'Direction Fields and Isoclines'

Thread 'Direction Fields and Isoclines'

I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...
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