1. **Problem Statement:** Solve the partial differential equation (PDE) $$\frac{\partial^2 z}{\partial y \partial x} = x^4 y$$ with initial conditions $$z(x,0) = x^4$$ and $$z(1,y) = \cos y$$. 2. **Recall the PDE and initial conditions:** $$\frac{\partial^2 z}{\partial y \partial x} = x^4 y$$ $$z(x,0) = x^4$$ $$z(1,y) = \cos y$$ 3. **Step 1: Integrate the PDE with respect to $y$ first.** Since $$\frac{\partial^2 z}{\partial y \partial x} = \frac{\partial}{\partial y} \left( \frac{\partial z}{\partial x} \right) = x^4 y,$$ integrate both sides with respect to $y$: $$\int \frac{\partial}{\partial y} \left( \frac{\partial z}{\partial x} \right) dy = \int x^4 y dy$$ which gives $$\frac{\partial z}{\partial x} = \frac{x^4 y^2}{2} + C(x)$$ where $C(x)$ is an arbitrary function of $x$. 4. **Step 2: Integrate with respect to $x$ to find $z$.** Integrate: $$z = \int \left( \frac{x^4 y^2}{2} + C(x) \right) dx = \int \frac{x^4 y^2}{2} dx + \int C(x) dx$$ $$= \frac{y^2}{2} \int x^4 dx + \int C(x) dx = \frac{y^2}{2} \cdot \frac{x^5}{5} + D(x) + E(y)$$ where $D(x) = \int C(x) dx$ and $E(y)$ is an arbitrary function of $y$. So, $$z = \frac{x^5 y^2}{10} + D(x) + E(y)$$ 5. **Step 3: Use initial condition $z(x,0) = x^4$.** Substitute $y=0$: $$z(x,0) = \frac{x^5 \cdot 0^2}{10} + D(x) + E(0) = D(x) + E(0) = x^4$$ Let $E(0) = k$ (constant), then $$D(x) = x^4 - k$$ 6. **Step 4: Use initial condition $z(1,y) = \cos y$.** Substitute $x=1$: $$z(1,y) = \frac{1^5 y^2}{10} + D(1) + E(y) = \frac{y^2}{10} + D(1) + E(y) = \cos y$$ Recall from step 5: $$D(1) = 1^4 - k = 1 - k$$ So, $$\frac{y^2}{10} + 1 - k + E(y) = \cos y$$ Rearranged: $$E(y) = \cos y - \frac{y^2}{10} - 1 + k$$ 7. **Step 5: Write the full solution.** Recall $$z = \frac{x^5 y^2}{10} + D(x) + E(y) = \frac{x^5 y^2}{10} + x^4 - k + \cos y - \frac{y^2}{10} - 1 + k$$ Simplify by canceling $-k$ and $+k$: $$z = \frac{x^5 y^2}{10} + x^4 + \cos y - \frac{y^2}{10} - 1$$ 8. **Final answer:** $$\boxed{z = \frac{x^5 y^2}{10} + x^4 + \cos y - \frac{y^2}{10} - 1}$$ This satisfies the PDE and the given initial conditions.