Mathematics I20805 marksNumerical: verify differential equation solutionIntroduction to first order equations Separable equations
Show that y = x - frac1x is a solution of the differential equation xy' + y = 2x.
Show that $y = x - \frac{1}{x}$ is a solution of the differential equation $xy' + y = 2x$. [5]
- Proposed solution: $y = x - \dfrac{1}{x}$ - Differential equation: $xy' + y = 2x$ --- $$y = x - \frac{1}{x} = x - x^{-1}$$ $$y' = \frac{d}{dx}(x) - \frac{d}{dx}(x^{-1}) = 1 - (-1)x^{-2} = 1 + \frac{1}{x^2}$$ --- $$xy' = x\left(1 + \frac{1}{x^2}\right) = x + \frac{x}{x^2} = x + \frac{1}{x}$$ ---...