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Problem 29 Hard Difficulty

Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen.
$ x^2 + 2y^2 = k^2 $

Answer

$$y=\pm x^{2} \cdot e^{C_{1}}=C x^{2}$$

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Video Transcript

this question asked us to find the orthogonal trajectories and then growth. Okay, we're looking at X squared plus two. Why Squared is casework Now? First off, we know Kay is a constant. Therefore, this is simply gonna be able to hear. Now differentiate this part Two acts plus four. Why do you? Why, Over detox is equivalent to zero. We know this is why Squared times two for why? And then we obviously have d y over Jack's, right? This just in terms of d Y over DX. We get negative X over two. Why, No, we know what we're gonna have is we need to look at the control of D X over acts, so DX divide by axe. This is equivalent to you don't have toe move things around again to get the wise on one side of the ass is on one side. D y over to what? This is a corpulent too natural log of X squared. Time's not reward f k or see whatever your constant is that you would like to know touch the absolute value of why it's absolute value given the l. A. Now we know we can cancel these off and enough with X squared is plus or minus Okay, times Why? Once you took with Ellen, you also take away the absolute value signs and then graphing this.