the graph below shows solution curves of the differential equation dy/dx = 1 - 8/3 x^5/3. find an equation…

Explanation:

Step1: Integrate the differential equation

We have $\frac{dy}{dx}=1 - \frac{8}{3}x^{5/3}$. Integrating both sides with respect to $x$, we get $y=\int\left(1-\frac{8}{3}x^{5/3}\right)dx$. Using the power - rule of integration $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $y=x-\frac{8}{3}\times\frac{x^{\frac{5}{3}+1}}{\frac{5}{3}+1}+C=x-\frac{8}{3}\times\frac{x^{\frac{8}{3}}}{\frac{8}{3}}+C=x - x^{\frac{8}{3}}+C$.

Step2: Use the given point to find C

The curve passes through the point $(1,0.5)$. Substitute $x = 1$ and $y=0.5$ into the equation $y=x - x^{\frac{8}{3}}+C$. So, $0.5=1-1 + C$. Solving for $C$, we find $C = 0.5$.

Answer:

$y=x - x^{\frac{8}{3}}+0.5$