1. **State the problem:** We are given velocity components for an incompressible, steady flow with constant viscosity:\ $$u(y)=\frac{U}{h}y - \frac{hy}{2\mu}\frac{dp}{dx}\left(1-\frac{y}{h}\right), \quad v=0, \quad w=0,$$\nand pressure $p=p(x)$, with constants $U, h,$ and $\frac{dp}{dx}$. We need to show that these satisfy the equation of motion without body forces.\ \n2. **Relevant equation:** The Navier-Stokes equation for steady incompressible flow without body forces in the $x$-direction is: $$0 = -\frac{1}{\rho}\frac{dp}{dx} + \nu \frac{d^2 u}{dy^2}$$ where $\nu = \frac{\mu}{\rho}$ is kinematic viscosity. The flow is fully developed so velocity depends only on $y$, and pressure depends only on $x$.\ \n3. **Calculate derivatives:**\ - First derivative of $u(y)$ with respect to $y$:\ $$\frac{du}{dy} = \frac{U}{h} - \frac{h}{2\mu}\frac{dp}{dx}\left(1-\frac{y}{h}\right) - \frac{hy}{2\mu}\frac{dp}{dx}\left(-\frac{1}{h}\right)$$\ Simplify the second term: $$ - \frac{h}{2\mu}\frac{dp}{dx} + \frac{y}{2\mu}\frac{dp}{dx}$$\ So, $$\frac{du}{dy} = \frac{U}{h} - \frac{h}{2\mu}\frac{dp}{dx} + \frac{y}{2\mu}\frac{dp}{dx} + \frac{y}{2\mu}\frac{dp}{dx} = \frac{U}{h} - \frac{h}{2\mu}\frac{dp}{dx} + \frac{y}{\mu}\frac{dp}{dx}$$\ \n4. **Second derivative of $u(y)$:**\ $$\frac{d^2 u}{dy^2} = \frac{d}{dy}\left(\frac{U}{h} - \frac{h}{2\mu}\frac{dp}{dx} + \frac{y}{\mu}\frac{dp}{dx}\right) = 0 + 0 + \frac{1}{\mu}\frac{dp}{dx} = \frac{1}{\mu}\frac{dp}{dx}$$\ \n5. **Substitute into momentum equation:** Multiply both sides of equation by $\rho$, rearranged as: $$0 = -\frac{dp}{dx} + \mu \frac{d^2 u}{dy^2}.$$ Substitute the second derivative value: $$-\frac{dp}{dx} + \mu \times \frac{1}{\mu}\frac{dp}{dx} = -\frac{dp}{dx} + \frac{dp}{dx} = 0.$$\ \n6. **Conclusion:** The given velocity profile satisfies the $x$-momentum equation for steady incompressible flow without body forces. The $v=0$ and $w=0$ components trivially satisfy their respective equations as there is no velocity component in those directions or pressure variation in $y$ or $z$.