Key Concepts and Formulas

• Linear First-Order Differential Equation: A differential equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$, where $P(x)$ and $Q(x)$ are functions of $x$.
• Integrating Factor (I.F.): For a linear first-order differential equation, the integrating factor is given by $I.F. = e^{\int P(x) dx}$. The solution is then $y(x) \cdot I.F. = \int Q(x) \cdot I.F. \ dx + C$.
• Trigonometric Identities: $\frac{1}{\sin y} = \csc y$ and $\frac{\cos y}{\sin y} = \cot y$.

Step-by-Step Solution

Step 1: Rewrite the given differential equation.

We are given: $(7x^4 \cot y - e^x \csc y) \frac{dx}{dy} = x^5$ Rewrite this as: $\frac{dy}{dx} = \frac{7x^4 \cot y - e^x \csc y}{x^5}$ $\frac{dy}{dx} = \frac{7x^4 \frac{\cos y}{\sin y} - e^x \frac{1}{\sin y}}{x^5}$ $\frac{dy}{dx} = \frac{7 \cos y}{x \sin y} - \frac{e^x}{x^5 \sin y}$

Step 2: Transform the equation.

Multiply both sides by $\sin y$: $\sin y \frac{dy}{dx} = \frac{7 \cos y}{x} - \frac{e^x}{x^5}$ Rearrange the terms: $\sin y \frac{dy}{dx} - \frac{7}{x} \cos y = - \frac{e^x}{x^5}$

Step 3: Use a suitable substitution.

Let $v = \cos y$. Then, $\frac{dv}{dx} = -\sin y \frac{dy}{dx}$. Substituting into the equation above, we get: $-\frac{dv}{dx} - \frac{7}{x} v = -\frac{e^x}{x^5}$ Multiply by -1: $\frac{dv}{dx} + \frac{7}{x} v = \frac{e^x}{x^5}$ This is now a linear first-order differential equation in terms of $v$ and $x$.

Step 4: Calculate the integrating factor.

The integrating factor (I.F.) is given by $I.F. = e^{\int P(x) dx}$, where $P(x) = \frac{7}{x}$. $I.F. = e^{\int \frac{7}{x} dx} = e^{7 \ln x} = e^{\ln x^7} = x^7$

Step 5: Find the general solution.

The general solution is given by: $v \cdot (I.F.) = \int Q(x) \cdot (I.F.) dx + C$ where $Q(x) = \frac{e^x}{x^5}$. So, $v \cdot x^7 = \int \frac{e^x}{x^5} \cdot x^7 dx + C$ $v x^7 = \int x^2 e^x dx + C$ We need to integrate $x^2 e^x$. Using integration by parts twice: $\int x^2 e^x dx = x^2 e^x - \int 2x e^x dx = x^2 e^x - 2(x e^x - \int e^x dx) = x^2 e^x - 2x e^x + 2e^x + C_1 = e^x(x^2 - 2x + 2) + C_1$ So, $v x^7 = e^x(x^2 - 2x + 2) + C$ Substituting back $v = \cos y$: $(\cos y) x^7 = e^x(x^2 - 2x + 2) + C$

Step 6: Apply the initial condition to find C.

We are given that the curve passes through the point $(1, \frac{\pi}{2})$. So, when $x=1$, $y = \frac{\pi}{2}$. $\cos \left(\frac{\pi}{2}\right) (1)^7 = e^1(1^2 - 2(1) + 2) + C$ $0 = e(1 - 2 + 2) + C$ $0 = e + C$ $C = -e$

Step 7: Find the particular solution.

Substituting $C = -e$ into the general solution: $(\cos y) x^7 = e^x(x^2 - 2x + 2) - e$

Step 8: Evaluate cos y at x=2.

We need to find $\cos y$ when $x=2$. $(\cos y) (2)^7 = e^2(2^2 - 2(2) + 2) - e$ $(\cos y) (128) = e^2(4 - 4 + 2) - e$ $128 \cos y = 2e^2 - e$ $\cos y = \frac{2e^2 - e}{128}$

Common Mistakes & Tips

• Remember to change the sign when substituting $v = \cos y$ to $-\frac{dv}{dx}$.
• Be careful with integration by parts and signs when evaluating the integral $\int x^2 e^x dx$.
• Don't forget to substitute back to the original variable $y$ after solving for $v$.

Summary

We solved the given differential equation by first rewriting it and then using the substitution $v = \cos y$ to transform it into a linear first-order differential equation. We found the integrating factor and general solution, then used the initial condition to find the particular solution. Finally, we evaluated $\cos y$ at $x=2$ to obtain the answer.

The final answer is $\boxed{\frac{2e^2 - e}{128}}$, which corresponds to option (C).