Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Experience the convenience of getting accurate answers to your questions from a dedicated community of professionals. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
[tex](x^2 - y^2) \, dx + 2xy \, dy = 0[/tex]
Multiply both sides by [tex]\frac1{x^2}[/tex].
[tex]\left(1 - \dfrac{y^2}{x^2}\right) \, dx + \dfrac{2y}x \, dy = 0[/tex]
Substitute [tex]y=vx[/tex], so [tex]v=\frac yx[/tex] and [tex]dy=x\,dv+v\,dx[/tex].
[tex](1-v^2) \, dx + 2v (x\,dv + v\,dx) = 0[/tex]
[tex](1 + v^2) \, dx + 2xv \, dv = 0[/tex]
Separate the variables.
[tex]2xv\,dv = -(1 + v^2) \, dx[/tex]
[tex]\dfrac{v}{1+v^2}\,dv = -\dfrac{dx}{2x}[/tex]
Integrate both sides
[tex]\displaystyle \int \frac{v}{1+v^2}\,dv = -\frac12 \int \frac{dx}x[/tex]
On the left side, substitute [tex]w=1+v^2[/tex] and [tex]dw=2v\,dv[/tex].
[tex]\displaystyle \frac12 \int \frac{dw}w = -\frac12 \int\frac{dx}x[/tex]
[tex]\displaystyle \ln|w| = -\ln|x| + C[/tex]
Solve for [tex]w[/tex], then [tex]v[/tex], then [tex]y[/tex].
[tex]e^{\ln|w|} = e^{-\ln|x| + C}[/tex]
[tex]w = e^C e^{\ln|x^{-1}|}[/tex]
[tex]w = Cx^{-1}[/tex]
[tex]1 + v^2 = Cx^{-1}[/tex]
[tex]1 + \dfrac{y^2}{x^2} = Cx^{-1}[/tex]
[tex]\implies \boxed{x^2 + y^2 = Cx}[/tex]
Your mistake is in the first image, between third and second lines from the bottom. (It may not be the only one, it's the first one that matters.)
You incorrectly combine the fractions on the left side.
[tex]\dfrac1{-2v} -\dfrac v{-2} = \dfrac1{-2v} - \dfrac{v^2}{-2v} = \dfrac{1-v^2}{-2v} = \dfrac{v^2-1}{2v}[/tex]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.