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Find an equation of the curve that passes through the point $ (0,2) $ and whose slope at $ (x, y) $ is $ x/y. $

$$y(0)=2>0$$

Differential Equations

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Oregon State University

Baylor University

University of Nottingham

Idaho State University

this question asked us for the equation of the curve that passes through the 0.0 or two and his slope X y is X over. Why? Okay, now what we know is that given the fact that we have a slope at acts over why this means that D y o ver de acts this is slope notation in this problem can be written as equal to X over. Why now? We want all the white terms on the left hand side. So we're just gonna rearrange this and then all the ex terms on the right hand side because now we can We're easily integrate. Increased the expo number one divide by the new exponents. Now, how why Squared is X squared plus two c not substituting X's here, wise to we know we now need to solve for C so we should get C equals and we end up with C equals two. Okay, Now substitute this about this value back into our equation plus two c So, plus two times two when they get why squared is X squared plus four. Now remember, we don't want y squared equals. We want y equals We must take the square root. Now notice how it should be positive or negative or plus or minus because we're doing the square root. However, remember why zero is, too, Which means this has to be positive, which means be just disregard the negative solutions. So this is the answer.