Solve this 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 this differential equation. We're given that x dy minus y squared minus 4y times dx equals 0, with the initial condition y of 1 equals 2, and x is greater than 0. First, let's rearrange the equation to separate the variables. We get x dy equals y squared minus 4y times dx. Dividing both sides by x and y squared minus 4y, we get dy over y squared minus 4y equals dx over x. This is now in a form where we can integrate both sides.
To solve this differential equation, we need to integrate both sides. The right side gives us log of x. For the left side, we first need to factor the denominator as y times y minus 4. Then we use partial fractions to decompose it. We get A over y plus B over y minus 4. Using standard methods, we find that A equals negative one-fourth and B equals one-fourth. Now we can integrate each term separately.
After integration, we get negative one-fourth log of absolute y, plus one-fourth log of absolute y minus 4, equals log of absolute x plus a constant. Using logarithm properties, we can rewrite this as log of the ratio of y minus 4 to the power of one-fourth, over y to the power of one-fourth, equals log of x plus log of K. Taking the exponential of both sides, we get that this ratio equals K times x. This is our general solution. Now, we need to use our initial condition to find K.
Now let's find the value of 10 y of square root of 2. From our solution, we have y minus 4 to the power of one-fourth equals K times x times y to the power one-fourth. Using the initial condition y of 1 equals 2, we get K equals negative 2 to the power one-fourth divided by 2 to the power one-fourth. Substituting this back, we raise both sides to the power 4 to get y minus 4 equals negative x to the power 4 times y. Rearranging, we get y times 1 plus x to the power 4 equals 4, or y equals 4 divided by 1 plus x to the power 4. When x equals square root of 2, x to the power 4 equals 4, so y of square root of 2 equals 4 divided by 5. Therefore, 10 times y of square root of 2 equals 8.
To solve this differential equation, we need to integrate both sides. The right side gives us log of x. For the left side, we first need to factor the denominator as y times y minus 4. Then we use partial fractions to decompose it. We get A over y plus B over y minus 4. Using standard methods, we find that A equals negative one-fourth and B equals one-fourth. Now we can integrate each term separately.
After integration, we get negative one-fourth log of absolute y, plus one-fourth log of absolute y minus 4, equals log of absolute x plus a constant. Using logarithm properties, we can rewrite this as log of the ratio of y minus 4 to the power of one-fourth, over y to the power of one-fourth, equals log of x plus log of K. Taking the exponential of both sides, we get that this ratio equals K times x. This is our general solution. Now, we need to use our initial condition to find K.
Now let's find the value of 10 y of square root of 2. From our solution, we have y minus 4 to the power of one-fourth equals K times x times y to the power one-fourth. Using the initial condition y of 1 equals 2, we get K equals negative 2 to the power one-fourth divided by 2 to the power one-fourth. Substituting this back, we raise both sides to the power 4 to get y minus 4 equals negative x to the power 4 times y. Rearranging, we get y times 1 plus x to the power 4 equals 4, or y equals 4 divided by 1 plus x to the power 4. When x equals square root of 2, x to the power 4 equals 4, so y of square root of 2 equals 4 divided by 5. Therefore, 10 times y of square root of 2 equals 8.
To summarize what we've learned: We solved the differential equation x dy minus y squared minus 4y dx equals 0 using separation of variables. We used partial fraction decomposition to integrate one over y squared minus 4y. After applying the initial condition y of 1 equals 2, we found the general solution to be y equals 4 divided by 1 plus x to the power 4. At x equals square root of 2, y equals 4/5, so 10 times y of square root of 2 equals 8. This approach can be applied to many first-order differential equations.