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Determine by inspection at least two solutions of the given first-order IVP. $$x y^{\prime}=2 y, \quad y(0)=0$$

Hi there we are given a differential equation, Y times Y plus X times y. Prime is equal to two Y and y off zero is equal to zero. So these are given. So we have to test um some Y functions. So the first one let's try um What if y is equal to zero then why prime? So also equal to zero? Let's let's test this is true for the differential equation. So if I'm gonna plug in so X times plug in Y prime at 00 which is equal to two times the value of Y in this case is zero. So this case is zero and the other part is also zero so zero is equal to zero. That holds. That means yes this is one solution of this differential equation. So but any question it says at least two solutions. So we have to find another solution here. So let's try another one. Let's say the second one. Um Let's say y is equal to X. And the y. Prime should be one. Let's plug in these values here. So X times one. Is it equal or not? Two times the value of X which is two X. So in this case two X is equal to um I mean X is equal to two X. So this is not true. That means this not holds for the solution. So if you look at the expression so we can just define another function for why? Let's say if I just say y is equal to X squared and y. Prime should be two X. So I'm gonna plug in these values to the equation. So X times Y prime two X. Is it equal to or not two times X squared. So when I just look at the equation this is two X squared and two X squared. Yeah, that works too. So the number three is the solution and number one is the solution here. So we got two solutions for this equation, but this is the answer of the question.

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