Question

If x^(h)y^(k) is the integrating factor of the following equation y^(2)(y dx+2xdy )+x^(2)(-2y dx-xdy )=0 then the values of h an k are

Answer 1

The values of ( h ) and ( k ) for the integrating factor
`\( x^h y^k \)` are ( h = -2 ) and ( k = 1 ).

Step-by-step explanation:

The given differential equation can be expressed as
`\(y^2(y \, dx + 2x \, dy) + x^2(-2y \, dx - x \, dy) = 0\)`. To find the integrating factor
`\( x^h y^k \)`, we need to compare the coefficients of (dx) and (dy) terms and solve for (h) and (k). Comparing the (dx) terms, we get (h = -2), and comparing the (dy) terms, we get (k = 1). Therefore, the integrating factor is
`\(x^(-2)y\)`.

The integrating factor is a function that helps transform a differential equation into an exact differential equation, making it easier to solve. In this case, the integrating factor
`\(x^(-2)y\)` is obtained by comparing the coefficients of the given differential equation. By multiplying the entire equation by this integrating factor, we can make it exact, facilitating the integration process.

Solving differential equations often involves identifying suitable integrating factors to simplify the equation. The values of (h) and (k) are determined by the coefficients in the given equation, and finding the integrating factor allows for a more straightforward solution to the differential equation.