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Problem 11 Easy Difficulty

Verify that the following functions are solutions to the given differential equation.
$y=\frac{1}{1-x}$ solves $y^{\prime}=y^{2}$

Answer

$y^{\prime}=\frac{d}{d x}\left(\frac{1}{1-x}\right)=\frac{1}{(1-x)^{2}} \qquad$ (reciprocal rule and chain rule)
$=y^{2} \quad$ (satisfying the differential equation)

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Video Transcript

we were asked to verify that Why? The function by equals to one over one minus X solves the differential equation by prime equals two y square. So we basically have to solve for y prime and show that it equals to the right hand side of the equation. Which is why square now, let's really use this derivative rule toe Find my prime duty eggs off any function and x rays to the power Are is our times for bex to the R minus one times of prime of X Here are fo Becks is going to be one minute six and our is gonna be negative warm. So by this rule, why prime? Um okay, By this rule, Y prime is going to be the duty over D X off one minus X rays to the negative one. So using this rule, it tickles Negative one one minute's X rays to the negative one minus one times the derivative off one minus x So this equals negative one times one minus X rays to the negative too. And derivative off one minus x will be negative one so negative one multiplied. With that negative one becomes the one and this simplifies toe one over one minus X square, which equals y squared. So we have shown that why prime equals Why square so dysfunction Why equals one over one minutes, ext solves the differential equation by prime equals y squared.