Question

Find the general solution to the differential equation

x²-4xy + xdy/dx=0
Find the integrating factor, u(x)=
Find y(x)=
Use C as the unknown constant.

Answer 1

To find the general solution to the differential equation x²-4xy + xdy/dx=0, you can first rearrange the equation and then separate the variables and integrate both sides. The general solution is y = ±(x^4 / C), where C is the unknown constant.

Step-by-step explanation:

To find the general solution to the differential equation x²-4xy + xdy/dx=0, we can first rearrange the equation:

x²-4xy + xdy/dx=0

After rearranging, we have:

x(dy/dx) - 4xy + x² = 0

This is a separable differential equation. To solve it, we can separate the variables and integrate both sides:

(dy/dx) = (4y - x)/(x)

By separating the variables and integrating, we get:

ln|y| = 4ln|x| - ln|C|

Now, we can exponentiate both sides to eliminate the natural logarithms:

|y| = |x|^4 / |C|

Finally, we can rewrite the absolute value expressions as:

y = ±(x^4 / C)

So, the general solution to the differential equation is y = ±(x^4 / C), where C is the unknown constant.