Question

Differential Equations. MEDAL!!!

Answer 1

if the slope of a curve y=f(x) is
\[\frac{ x+y }{ -x }\]
at all points. given the point (1,3)
find the equation of the curve

Answer 2

\[f'(x)=y'=-\dfrac{x+y}{x}\]

\[-y'x=-\frac{dy}{dx}x=x+y\implies -\frac{dy}{dx}=1+\dfrac{y}{x}\]
\[tx=y\]
\[-(\dfrac{dt}{dx}x+t)=1+t\]
\[\dfrac{dt}{dx}x=-2t\implies\dfrac{dt}{2t}=-\dfrac{dx}{x}\]

\[\ln 2t=\ln 2(\dfrac{y}{x})=\ln x+c\]
plug in (1,3) to get c

Answer 3

thanks a lot

Answer 4

\[\frac{ dy }{dx }=-\frac{ x+y }{x}=-1-\frac{ y }{x }\]
\[\frac{ dy }{dx }+\frac{ 1 }{ x }y=-1\]
\[I.F=e ^{\int\limits \frac{ 1 }{ x }dx}=e ^{\ln x}=x\]
c.s. is \[y*x=\int\limits -x~dx=\frac{ -x^2 }{2 }+c\]
when x=1,y=3
\[3*1=-\frac{ 1^2 }{ 2 }+c\]
\[or~c=3+\frac{ 1 }{2 }=\frac{ 7 }{2 }\]
hence eq. of curve is \[yx=\frac{ -x^2 }{2 }+\frac{ 7 }{2 }\]
\[or~2xy=7-x^2\]