Question

Solve the given differential equation by separation of variables: y ln(x) dx dy = y 1 x 2?

Answer 1

The differential equation is solved by separating variables, integrating each side, and then solving for y in terms of x to find the general solution.

Step-by-step explanation:

Solution to Differential Equation

To solve the differential equation by separation of variables, we begin by rewriting the given equation as:

y ln(x) dx = y dy / x2

We then separate the variables by dividing both sides by y and multiplying by x2:

(1/y) dy = (x ln(x) / x2) dx

Integrating both sides:

1. 
   
3. ∫ (1/y) dy = ln(y) + C1
4. 
   
6. ∫ (x ln(x) / x2) dx = ∫ (ln(x)/x) dx = (1/2)(ln(x))2 + C2

Equating the two sides and solving for y, we can find the general solution in terms of x.

Note that constants of integration C1 and C2 can be combined into a single constant C.