Question

Solve the differential equation (x-4e^y -9)dx +(4x+e^y - 2)e^y dy=0. Find the general solution.

a) y = ln(|x + 2e^y - 3|)
b) y = ln(|x - 2e^y + 3|)
c) y = ln(|x - 2e^y - 3|)
d) y = ln(|x + 2e^y + 3|)

Answer 1

The general solution to the given differential equation is:
`(1/2)x^2 - 2xe^y - 9x + (1/4)e^(^2^y^) - 2e^y = C`

None of the given options is correct

Explanation:

To solve the given differential equation, we can separate the variables and integrate both sides.

The given differential equation is:
`(x - 4e^y - 9)dx + (4x + e^y - 2)e^y dy = 0`

To separate the variables, we can move all the terms involving x to the left side and all the terms involving y to the right side:

`(x - 4e^y - 9)dx = -(4x + e^y - 2)e^y dy`

Next, we integrate both sides:

∫(x - 4e^y - 9)dx = -∫(4x + e^y - 2)e^y dy

Integrating the left side with respect to x gives:

`(1/2)x^2 - 4xe^y - 9x + C1`

Integrating the right side with respect to y involves some algebraic manipulation. We expand the expression and then integrate each term separately:

`- \int\limits (4x + e^y - 2)e^y dy\\ = - \int\limits (4xe^y + e^(2^y^) - 2e^y) dy\\ = -\int\limits4xe^y dy - \int\limits e^(2^y^) dy + 2∫e^y dy\\= -4\int\limitsxe^y dy - (1/2)e^(2^y^) + 2e^y + C2`

Combining the results from both sides, we have:

`(1/2)x^2 - 4xe^y - 9x + C1 = -4∫xe^y dy - (1/2)e^(2^y^) + 2e^y + C2`

To find the general solution, we can combine the constants of integration:

`(1/2)x^2 - 4xe^y - 9x + C1 = -4∫xe^y dy - (1/2)e^(2^y^) + 2e^y + C2`

Simplifying the expression, we obtain:

`(1/2)x^2 - 4xe^y - 9x + C1 = -2xe^y - (1/4)e^(2^y^) + 2e^y + C2`

Rearranging the terms and combining constants, we have:

`(1/2)x^2 - 4xe^y + 2xe^y - 9x + (1/4)e^(2^y^) - 2e^y = C1 - C2`

Further simplifying the expression, we get:

`(1/2)x^2 - 2xe^y - 9x + (1/4)e^(^2^y^) - 2e^y = C`

Where C represents the combined constants of integration.

Therefore, the general solution to the given differential equation is:

`(1/2)x^2 - 2xe^y - 9x + (1/4)e^(^2^y^) - 2e^y = C`

None of the given options is correct