Homogeneous differential equation

kimkibun
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how am i going to determine if a higher order differential equation is homogenous? example,

d4y/dx4+d2y/dx2=y

d3y/dx3-d2y/dx2=0
 
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Homogeneous means all terms depend on y, no matter the order of the ODE. In your examples, both equations would be homogeneous. If you would add a term like 1 or f(x) it would become nonhomogeneous, i.e. y"=y is homogeneous, but y"=y+f(x) is nonhomogeneous.
 
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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