Solve the given question---Q.2 If y(x) is the solution of the differential equation $x dy - (y^2 - 4y) dx = 0$ for $x > 0$, $y(1) = 2$, and the slope of the curve $y = y(x)$ is never zero, then the value of $10 y(\sqrt{2})$ is ______.

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Let's solve the differential equation x dy minus y squared minus 4y dx equals 0, with the initial condition y of 1 equals 2, and x greater than 0. First, we'll rewrite this as x times dy/dx equals y squared minus 4y, or dy/dx equals (y squared minus 4y) divided by x. Next, we separate the variables to get dy over y squared minus 4y equals dx over x. We can rewrite the left side as dy over y times (y minus 4). The direction field shows the slope at different points, with our initial condition at the point (1,2). To solve this differential equation, we'll continue by using partial fractions. For the term 1 over y times (y minus 4), we decompose it as A over y plus B over (y minus 4). After finding the coefficients A equals negative 1/4 and B equals 1/4, we integrate both sides. On the right side, we get the natural logarithm of x plus a constant. After simplifying and solving for y, we get y equals 4 divided by (1 plus x to the fourth power). Using our initial condition y(1) equals 2, we find that y of square root of 2 equals 4/5. Therefore, 10 times y of square root of 2 equals 8. To summarize what we've learned: We solved the differential equation using separation of variables and partial fractions. The key insight was recognizing how to decompose the term one over y times y minus 4. After integrating both sides and applying the initial condition, we found that the solution is y equals 4 divided by 1 plus x to the fourth power. This allowed us to calculate that 10 times y of square root of 2 equals 8, which is our final answer. Now we'll use partial fractions to decompose the term 1 over y times y minus 4. We get A over y plus B over y minus 4. Setting y equals 0, we find A equals negative 1/4. Setting y equals 4, we find B equals 1/4. Next, we integrate both sides of our equation. For the left side, we get 1/4 times the natural logarithm of the absolute value of y minus 4 over y. For the right side, we get the natural logarithm of x plus a constant. After simplifying, we can express this as y minus 4 over y equals C times x to the fourth power. The slope at our initial point is shown by the red dashed line, and the blue curve represents our partial solution. Now we'll apply the initial condition y of 1 equals 2 to find the constant C. Substituting x equals 1 and y equals 2 into our equation, we get negative 1 equals C. This gives us the particular solution y minus 4 over y equals negative x to the fourth power. Solving for y, we get y equals 4 divided by 1 plus x to the fourth power. To find y of square root of 2, we substitute x equals square root of 2 into our solution. This gives us y of square root of 2 equals 4 divided by 5. Therefore, 10 times y of square root of 2 equals 8, which is our final answer. The red curve shows our solution, with the initial point in blue and our target point at square root of 2 in red. Let's verify that our solution satisfies the condition that the slope is never zero. From the differential equation, we know that the slope is given by y squared minus 4y divided by x. This slope is zero when y squared minus 4y equals zero, which means y equals 0 or y equals 4. Our solution y equals 4 divided by 1 plus x to the fourth power is never equal to 0 for x greater than 0. And it equals 4 only when x equals 0, which is outside our domain since we're told x is greater than 0. Therefore, the slope is never zero for our solution, confirming that our approach was valid. The green tangent lines on our graph illustrate that the slope is always negative. Our final answer is 10 times y of square root of 2 equals 8. To summarize what we've learned: We solved the differential equation using separation of variables and partial fractions. The key insight was recognizing how to decompose the term one over y times y minus 4. After integrating both sides and applying the initial condition, we found that the solution is y equals 4 divided by 1 plus x to the fourth power. This allowed us to calculate that 10 times y of square root of 2 equals 8, which is our final answer. The solution curve never crosses y equals 0 or y equals 4 for x greater than 0, confirming that our approach was valid.