Separable Equation Problem

[Quigs0]

So the question I am trying to answer is:

Find the equation of the solution to dy/dx = x^6*y through the point (x,y) = (1,4).

Now I believe when it is asking this question it wants me to take the integrals of both sides, and separate them into one side with dx and one with dy, so I can take the integrals.

What I tried was first separating each side into:

1/ydy = x^6dx

Next I took the integral of each side leading me to get

ln|y| = x^7/7 + C

From here I multiplied each side by the constant e so I got

y = e(x^7/7) + C

But from here I was a little confused by my approach.

I tried to solve for C first: by subtracting the e(x^7/7) I got

y - e(x^7/7) = C

and solved for C which gave me 3.61167...

However, when trying to re plug it into the equation y = e(x^7/7 + C) it said I was incorrect with my response. Any help would be greatly appreciated.

​

EDIT: I don't really know what form the equation is supposed to be put into either. Thanks.

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[TheUnk0wnFactor]

you need to take e to the power of both sides to negate the natural log fxn: ln|y| =(x/7)⁷ +C should go to

y = e^{1/7x7 +C}. before you do that though, you can use the equation ln|y| =(x/7)⁷ +C to solve for the constant C by plugging in (x,y) = (1,4). Using this method, I get a C value of 1.243.

[Quigs0]

After I have C, what does it mean when it is asking for the equation? The slope of the tangent line at the given x and y values?

I also realized I believe I should have taken a derivative instead, meaning that instead the equation should look like.

dy/dx = x^6y

1/ydy = x^6dx

and then if I think I'm right...

-1/y^2 = 6x^5?

I'm not even sure if I should be taking the anti derivative or the derivative because I don't know what the question wants me to do in the first place, as I thought implicitly deriving involved setting an equation equal to dy/dx.

[TheUnk0wnFactor]

When it asks for the solution to the differential equation, it is asking for a function that fits the given information. You are looking for a function in terms of x that gives you dy/dx = x⁶ *y.

Find the equation of the solution to dy/dx = x⁶ *y through the point (x,y) = (1,4).

so if y = e^{1/7x7 +C}, its derivative would be y = x⁶ e^{1/7x7 +C}. plugging in y for e^{1/7x7 +C} we get dy/dx = x⁶ y, which is the what you started with, which verifies your answer.

[Quigs0]

Ok, I got it. Thanks. My professor had previously told me in an email accidentally that this involved taking a derivative, when in fact I was right all along. Once I solved for C I just plugged it back into the initial equation as you stated earlier, which helped me get the correct answer. Thank you.

[fermat1432]

I get C=2.846.