solve the equation.\n\frac{dy}{dx}=\frac{x}{7cos(y)}\nchoose 1 answer:\n(a) y = \frac{cos(x)+xsin(x)}{7cos^{2…

Explanation:

Step1: Separate variables

Separate the variables in the differential - equation $\frac{dy}{dx}=\frac{x}{7\cos(y)}$. We get $7\cos(y)dy = xdx$.

Step2: Integrate both sides

Integrate the left - hand side $\int7\cos(y)dy$ and the right - hand side $\int xdx$. The integral of $7\cos(y)$ with respect to $y$ is $7\sin(y)$ (since $\int\cos(y)dy=\sin(y)$ and by the constant multiple rule of integration $\int kf(y)dy = k\int f(y)dy$ where $k = 7$), and the integral of $x$ with respect to $x$ is $\frac{x^{2}}{2}+C_1$ (using the power rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$, here $n = 1$). So, $7\sin(y)=\frac{x^{2}}{2}+C_1$.

Step3: Solve for $y$

First, rewrite the equation as $\sin(y)=\frac{x^{2}}{14}+\frac{C_1}{7}$. Let $C=\frac{C_1}{7}$, then $y=\arcsin(\frac{x^{2}}{14}+C)$.

Answer:

B. $y = \arcsin(\frac{x^{2}}{14}+C)$