If 𝑦(𝑥) is the solution of the differential equation 𝑥𝑑𝑦 − (𝑦2 − 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(√2 ) is

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In this problem, we need to solve the differential equation x dy minus y squared minus 4y dx equals 0, where x is greater than 0, and y of 1 equals 2. We're asked to find the value of 10 times y of square root of 2. Let's start by rewriting the equation in standard form. We can rearrange to get x times dy/dx equals y squared minus 4y, which gives us dy/dx equals (y squared minus 4y) divided by x. Next, we'll separate the variables to prepare for integration. We get dy over y squared minus 4y equals dx over x. We can also rewrite the left side as dy over y times (y minus 4). Now we need to integrate both sides of our equation. For the left side, we'll use partial fraction decomposition to break down 1 over y times (y minus 4). We can write this as A over y plus B over (y minus 4). To find the constants A and B, we set up the equation 1 equals A times (y minus 4) plus B times y. When y equals 0, we get 1 equals negative 4A, so A equals negative 1/4. When y equals 4, we get 1 equals 4B, so B equals 1/4. This gives us 1 over y times (y minus 4) equals 1/4 times (1 over (y minus 4) minus 1 over y). Now we can integrate both sides. The left side becomes 1/4 times (natural log of absolute value of (y minus 4) minus natural log of absolute value of y), which equals 1/4 times natural log of absolute value of (y minus 4) over y. The right side becomes natural log of absolute value of x plus a constant C1. After simplifying, we get the absolute value of (y minus 4) over y equals C times x to the power of 4, where C is a non-zero constant. Now we'll use the initial condition y of 1 equals 2 to find the particular solution. Substituting x equals 1 and y equals 2 into our general solution, we get (2 minus 4) over 2 equals C times 1 to the power of 4. This simplifies to negative 2 over 2 equals C, so C equals negative 1. Now we can write our particular solution. We have (y minus 4) over y equals negative x to the power of 4. Rearranging, we get 1 minus 4 over y equals negative x to the power of 4, which gives us 4 over y equals 1 plus x to the power of 4. Finally, solving for y, we get y equals 4 over (1 plus x to the power of 4). We should verify that the slope is never zero. The slope is given by dy/dx equals (y squared minus 4y) over x, which equals y times (y minus 4) over x. For our solution, y is always between 0 and 4 for x greater than 0, so y is never equal to 0 or 4. Therefore, the slope is never zero, as required. Now we can find the value of 10 times y of square root of 2. First, let's evaluate y of square root of 2 using our solution y equals 4 over (1 plus x to the power of 4). When x equals square root of 2, x to the power of 4 equals square root of 2 to the power of 4, which is 2 squared, or 4. So y of square root of 2 equals 4 over (1 plus 4), which is 4 over 5. Finally, we multiply by 10 to get 10 times y of square root of 2 equals 10 times 4 over 5, which equals 40 over 5, or 8. Therefore, the value of 10 times y of square root of 2 is 8. Let's summarize what we've learned. We solved the differential equation x dy minus y squared minus 4y dx equals 0, with the initial condition y of 1 equals 2. We approached this by separating variables and using partial fraction decomposition to integrate the left side. After applying the initial condition, we found that the solution is y equals 4 over 1 plus x to the power of 4. We verified that the slope of this curve is never zero, as required by the problem, by showing that y is always between 0 and 4 for x greater than 0. Finally, we calculated that 10 times y of square root of 2 equals 8, which is our answer.