- #1
TogoPogo
- 14
- 0
The problem states:
"By using the substitution y=xu, show that the differential equation \frac{dy}{dx}=\frac{y+\sqrt{x^{2}+y^{2}}}{x}, x>0 can be reduced to the d.e. x\frac{du}{dx}=\sqrt{u^{2}+1}.
Hence, show that if the curve passes through the point (1,0), the particular solution is given by y=\frac{1}{2}(x^{2}-1)."
I managed to get the d.e. into the form x\frac{du}{dx}=\sqrt{u^{2}+1} but I have no idea how to integrate \frac{du}{\sqrt{u^{2}+1}}. Wolfram Alpha is giving me some inverse hyperbolic sine stuff which I haven't learned yet (I'm in high school). All I've really 'learned' from my teacher so far was solving separable DE's, and inseparable DE's with y=ux, however some of the questions that we were given required other techniques like integrating factors and stuff. Is this DE a special case or something?
Anyways, how would I approach this? Do I square both sides to get rid of the square root sign?
Many thanks.
"By using the substitution y=xu, show that the differential equation \frac{dy}{dx}=\frac{y+\sqrt{x^{2}+y^{2}}}{x}, x>0 can be reduced to the d.e. x\frac{du}{dx}=\sqrt{u^{2}+1}.
Hence, show that if the curve passes through the point (1,0), the particular solution is given by y=\frac{1}{2}(x^{2}-1)."
I managed to get the d.e. into the form x\frac{du}{dx}=\sqrt{u^{2}+1} but I have no idea how to integrate \frac{du}{\sqrt{u^{2}+1}}. Wolfram Alpha is giving me some inverse hyperbolic sine stuff which I haven't learned yet (I'm in high school). All I've really 'learned' from my teacher so far was solving separable DE's, and inseparable DE's with y=ux, however some of the questions that we were given required other techniques like integrating factors and stuff. Is this DE a special case or something?
Anyways, how would I approach this? Do I square both sides to get rid of the square root sign?
Many thanks.
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