Solve Differential Equation of Family of Curves and Orthogonal Trajectories

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Doing some extra credit and got stuck on this one.


Find the differential equation of the family of curves and of the orthogonal trajectories.

y = c - 2x

Needing a little help on this one...

Thanks
 
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"The differential equation of a family of curves" means a differential equation such that that the general solution to the differential equation is that family of curves. Can you think of a differential equation that has
y= c- 2x as its general solution?

One curve is orthogonal to another if there tangent lines are perpendicular where they intersect. What does this tell you about there derivatives?

(Actually, all "curves" satisfying y= c-2x are straight lines.)
 
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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