Breaking down Question 1:

The problem asks us to verify if the expression $y^2 - 4x^2 + e^{xy} = C$ is a solution to the differential equation:

$(8x - y e^{xy}) - (2y + x e^{xy}) y' = 0$

Step-by-Step Breakdown

1. Differentiate the given solution expression with respect to $x$: To verify if $y^2 - 4x^2 + e^{xy} = C$ is indeed a solution, we start by differentiating both sides of the equation with respect to $x$: $\frac{d}{dx}(y^2 - 4x^2 + e^{xy}) = \frac{d}{dx}(C)$ Since $C$ is constant, $\frac{d}{dx}(C) = 0$.

2. Apply the Product Rule and Chain Rule: Differentiate each term on the left-hand side: $2y \cdot y' - 8x + (y + xy') e^{xy} = 0$ Rearrange to group terms involving $y'$: $(2y + x e^{xy}) y' + y e^{xy} - 8x = 0$

3. Isolate $y'$: Move terms to isolate $y'$: $(2y + x e^{xy}) y' = 8x - y e^{xy}$ So, $y' = \frac{8x - y e^{xy}}{2y + x e^{xy}}$

4. Substitute $y'$ back into the differential equation: Substitute this $y'$ expression into the original differential equation: $(8x - y e^{xy}) - (2y + x e^{xy}) \left(\frac{8x - y e^{xy}}{2y + x e^{xy}}\right) = 0$ Simplifying this expression shows that it holds true, confirming that $y^2 - 4x^2 + e^{xy} = C$ is indeed a solution of the differential equation.

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Related Questions

1. How would you verify if a different function is a solution to the same differential equation?
2. What steps are involved in isolating terms when solving for $y'$ in implicit differentiation?
3. How does the presence of exponential terms like $e^{xy}$ impact the solution process for differential equations?
4. What are alternative methods for confirming solutions to differential equations?
5. How does solving differential equations with constants, like $C$, differ from those without?

Tip:

When differentiating complex expressions, clearly separate and simplify terms involving derivatives to avoid errors in calculations.