$\left(x^{2}+1\right) y^{\prime}=x \cdot y \cdot\left(\cos (7 \cdot \pi / 2) \cdot \frac{y^{2}-7^{2}}{7 \cdot y}+\sin (7 \pi / 2) \cdot\left(1+\frac{x^{2}+1}{y}\right)\right), y(0)==$

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Evaluate definite integrals using the power rule

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\left(x^{2}+1\right) y^{\prime}=x \cdot y \cdot\left(\cos (7 \cdot \pi / 2) \cdot \frac{y^{2}-7^{2}}{7 \cdot y}+\sin (7 \pi / 2) \cdot\left(1+\frac{x^{2}+1}{y}\right)\right), y(0)==

Simplify Trigonometric Functions: First, let's simplify the trigonometric functions $\cos(\frac{7\pi}{2})$ and $\sin(\frac{7\pi}{2})$ . Since $\cos(\frac{7\pi}{2}) = 0$ and $\sin(\frac{7\pi}{2}) = -1$ , the equation simplifies to: $\newline$ $(x^{2}+1)y' = x\cdot y\cdot (-1\cdot(1+\frac{x^{2}+1}{y}))$ .
Distribute and Simplify: Now, let's distribute the $x*y$ on the right side and simplify: $\newline$ $(x^{2}+1)y' = -x*y - x*(x^{2}+1)$ .
Separate Variables and Integrate: We can now separate variables and integrate both sides. Move all $y$ terms to one side and $x$ terms to the other side: $\newline$ $\frac{y'}{y} = -x\frac{x^{2}+1}{x^{2}+1}$ .
Simplify Right Side: Simplify the right side of the equation: $\frac{y'}{y} = -x$ .
Integrate Both Sides: Now integrate both sides: $\int(\frac{1}{y})dy = \int(-x)dx$ .
Exponentiate to Solve for y: The integrals are: $\newline$ $\ln|y| = -\frac{x^{2}}{2} + C$ .
Find Value of C: Exponentiate both sides to solve for $y$ : $y = e^{(-x^{2}/2 + C)}.$
Find Value of C: Exponentiate both sides to solve for $y$ : $y = e^{(-x^{2}/2 + C)}.$ We can rewrite $C$ as $e^{C}$ to simplify the equation: $y = e^{C} \cdot e^{(-x^{2}/2)}.$
Find Value of C: Exponentiate both sides to solve for $y$ : $y = e^{(-x^{2}/2 + C)}.$ We can rewrite $C$ as $e^C$ to simplify the equation: $y = e^C \cdot e^{(-x^{2}/2)}.$ Let's use the initial condition $y(0)$ to find the value of $C$ . Plugging in $x = 0$ , we get: $y(0) = e^C \cdot e^0.$
Find Value of C: Exponentiate both sides to solve for $y$ : $y = e^{(-x^{2}/2 + C)}.$ We can rewrite $C$ as $e^{C}$ to simplify the equation: $y = e^{C} \cdot e^{(-x^{2}/2)}.$ Let's use the initial condition $y(0)$ to find the value of $C$ . Plugging in $x = 0$ , we get: $y(0) = e^{C} \cdot e^{0}.$ Since $e^{0} = 1$ , we have: $y(0) = e^{C}.$ But we don't have the value for $y(0)$ , so we can't find $C$ . The problem seems to be missing the initial condition value for $y(0)$ .