Question

(X-1)dy-(y^2-6y)dx=0

Answer 1

$(x-1)dy-(y^2-6y)dx=0;$

$\frac{dy}{(y^2-6y)}-\frac{dx}{x-1}=0;$

$\frac{dy}{y^2-6y} =\frac{dx}{x-1}$

$\int\frac{dy}{y^2-6y} =\int\frac{dx}{x-1} ;$

1) $\int\frac{dy}{y^2-6y}=\left \{ {{\frac{A}{y}+\frac{B}{y-6}=\frac{1}{y(y-6)} } \atop {A(y-6)+By=1}}\atop{B=\frac{1}{6} };A=-\frac{1}{6} \right.$   $\int\frac{dy}{y^2-6y}=\int-\frac{1}{6y}+\frac{1}{6(y-6)}dy=\frac{1}{6}\int\frac{1}{y-6}-\frac{1}{6}\int\frac{1}{y}= \frac{1}{6}ln|y-6|-\frac{1}{6}ln|y|=\frac{1}{6}ln|\frac{y-6}{y} |+C$

2) $\int\frac{dx}{x-1}=ln|x-1|+C$

$\frac{1}{6}ln|\frac{y-6}{y} |=ln|x-1|+C$

$ln|\frac{y-6}{y} |=6ln|xC-C|;$

$\frac{y-6}{y}=(xC-C)^{6};$

$y=\frac{6}{1-(xC-C)^6}$

$\boxed{y=\frac{6}{1-C(x-1)^6} }$

Answer 2

The given first-order differential equation is:

(X - 1)dy - (y^2 - 6y)dx = 0

This is a first-order, nonlinear, and separable differential equation. To solve it, we can rearrange the terms and separate the variables. Here's how you can do it:

1. Move the terms involving y to the left side and the terms involving x to the right side:

(X - 1)dy = (y^2 - 6y)dx

2. Divide both sides by (X - 1):

dy = (y^2 - 6y)dx / (X - 1)

3. Now, we can separate the variables by dividing both sides into a fraction of functions of y and functions of x:

dy / (y^2 - 6y) = dx / (X - 1)

4. Next, we can use partial fraction decomposition on the left side to make it easier to integrate:

dy / (y^2 - 6y) = dy / (y(y - 6))

We'll use partial fraction decomposition to express dy / (y(y - 6)) as a sum of two simpler fractions:

dy / (y(y - 6)) = A/y + B/(y - 6)

To find A and B, we can multiply both sides by y(y - 6) to clear the denominators:

1 = A(y - 6) + By

Now, we can solve for A and B. Let's set y = 0 to solve for A:

1 = A(0 - 6) + B(0) 1 = -6A A = -1/6

Now, set y = 6 to solve for B:

1 = A(6 - 6) + B(6) 1 = 6B B = 1/6

So, we have:

dy / (y^2 - 6y) = (-1/6)dy / y + (1/6)dy / (y - 6)

5. Now, we can integrate both sides:

∫(1/y - 1/(y - 6)) dy = ∫(1/(X - 1)) dx

6. Integrate each side separately:

ln|y| - ln|y - 6| = ln|X - 1| + C

7. Combine the natural logarithms on the left side using properties of logarithms:

ln|y / (y - 6)| = ln|X - 1| + C

8. Exponentiate both sides to remove the natural logarithms:

|y / (y - 6)| = |X - 1|e^C

9. Since C is an arbitrary constant, we can replace it with another constant, say, K:

|y / (y - 6)| = K|X - 1|

10. Finally, remove the absolute value signs:

y / (y - 6) = ±K(X - 1)

This is the general solution to the given differential equation. It's written with a ± sign because the absolute value signs can yield both positive and negative values. The sign choice will depend on the initial conditions or specific context of the problem.