Let's walk through the suggested solution for Q1 from the assignment:

You are asked to verify whether the expression $y^2 - 4x^2 + e^{xy} = C$ is a solution to the given ODE:
$(8x - y e^{xy}) - (2y + x e^{xy})y' = 0.$

Suggested Solution Breakdown:

1. Differentiate the given equation $y^2 - 4x^2 + e^{xy} = C$ with respect to $x$: $(y^2 - 4x^2 + e^{xy})' = (C)'.$ This gives: $2yy' - 8x + (y + x y')e^{xy} = 0.$ Simplifying further, we get: $2yy' + x y' e^{xy} - 8x + y e^{xy} = 0.$

2. Isolate $y'$ in the resulting equation: $2yy' + x y' e^{xy} = 8x - y e^{xy}.$ Solving for $y'$: $y' = \frac{8x - y e^{xy}}{2y + x e^{xy}}.$

3. Substitute $y'$ into the original ODE $(8x - y e^{xy}) - (2y + x e^{xy})y' = 0$ to verify the solution: Substituting $y' = \frac{8x - y e^{xy}}{2y + x e^{xy}}$ into the ODE, we get: $(8x - y e^{xy}) - (2y + x e^{xy})\frac{8x - y e^{xy}}{2y + x e^{xy}} = 0.$ Simplifying the right-hand side: $(8x - y e^{xy}) - (8x - y e^{xy}) = 0.$ Since this equation holds true, we confirm that $y^2 - 4x^2 + e^{xy} = C$ is indeed a solution to the given ODE.

Would you like more details or have any questions about specific steps?

Related questions to explore further:

1. How do you systematically differentiate implicit functions in similar cases?
2. Why is solving for $y'$ important when verifying a solution to an ODE?
3. Can this method be applied to non-linear differential equations in general?
4. What are other common techniques for verifying solutions to ODEs?
5. How does the presence of exponential terms affect solving differential equations?

Tip:

When differentiating, carefully handle exponential terms that involve products like $e^{xy}$, applying both product and chain rules effectively.