A curve which passes through the point (1,0) satisfies the equation dx/dy = x³
Find its equation
thanks for any help
Hello, gracey!
This is a Differential Equation problem . . .A curve which passes through the point (1,0)
satisfies the equation: .$\displaystyle \frac{dx}{dy} \:=\:x^3$
Find its equation
We have: .$\displaystyle \frac{dy}{dx} \:=\:x^{-3} \quad\Rightarrow\quad dy \:=\:x^{-3}\,dx$
Integrate: .$\displaystyle \int dy \;=\;\int x^{-3}\,dx \quad\Rightarrow\quad y \;=\;-\frac{x^{-2}}{2} + C \;=\;-\frac{1}{2x^2} + C$
Since (1,0) satisfies the equation: .$\displaystyle 0 \:=\:-\frac{1}{2\cdot1^2} + C \quad\Rightarrow\quad C \:=\:\frac{1}{2}$
Therefore, the equation is: .$\displaystyle y \;=\;-\frac{1}{2x^2} + \frac{1}{2}$