Question

Solve the separable differential equation 7x-4ysqrt(x^2+1)dy/dx=0 Subject to the initial condition: y(0)=3

Answer 1

\[3x-4y\sqrt{x^2+1}\frac{dy}{dx}=0 \]
Like that?

Answer 2

\[\frac{3x}{\sqrt{x^2+1}} dx=4ydy\]
Where do you get stuck here?

Answer 3

it took the previous question with a negative ivp... \[7x-4y \sqrt{x^2+1}\frac{ dy }{dx }=0 \] y(0)=3

Answer 4

Pardon me, not sure why I turned the \(7x\) into a \(3x\) in case the question didn't update ^^, however yes, it's separable and easy to integrate. The initial condition will only help you find the yet unknown constant.

Answer 5

i'm getting \[y(x)=\sqrt{\frac{ \sqrt{(3x \sqrt{x^2+1}+18)} }{ 2 }}\]

Answer 6

it's saying the answer isn't correct, I don't know where I went wrong

Answer 7

Lets look at the left hand side first, because it's the more complicated integral.

\[\Large \int \frac{7x}{\sqrt{x^2+1}}dx \]
this integral can easy be solved with an u substitution: u=x^2+1

\[\Large \frac{7}{2}\int \frac{1}{\sqrt{u}}du=7\sqrt{x^2+1} \]

Answer 8

and for the right hand side we get 2y^2

Answer 9

You can now put your initial condition on either side and solve for it.

Answer 10

\[\Large 7\sqrt{x^2+1}=2y^2+C \]

Answer 11

\[y(x)=\sqrt{\frac{ 7\sqrt{x^2+1}-7\sqrt{1}-18 }{ 2 }}\] is what i'm getting now

Answer 12

\[\Large 7\sqrt{x^2+1}=2y(x)^2+C \]

\[\Large y(0)=3 \]
\[\Large 7\sqrt{0^2+1}=2\cdot3^2+C \]
such that:
\[C=-11 \]

Answer 13

which should then give me \[y(x)=\sqrt{\frac{ 7\sqrt{x^2+1}+11 }{ 2 }}\] right?

Answer 14

yes

Answer 15

for some reason it's not accepting that as the answer, I worked it out a few times and I keep getting the same answer now, i'm hoping the answer was saved wrong in the system

Answer 16

http://www.wolframalpha.com/input/?i=7x-4ysqrt%28x%5E2%2B1%29dy%2Fdx%3D0%2C+y%280%29%3D3

Answer 17

It's right ;) well done

Answer 18

thank you!