2.8k views
2 votes
Solve the given differential equation by separation of variables.
eˣy dy/dx = e⁻ʸ + e⁻⁸ˣ⁻ʸ

User Matcheek
by
7.6k points

1 Answer

0 votes

Final answer:

To solve the given differential equation eˣy dy/dx = e⁻ʸ + e⁻⁸ˣ⁻ʸ by separation of variables, we need to separate the variables on each side of the equation and integrate.

Step-by-step explanation:

To solve the given differential equation eˣy dy/dx = e⁻ʸ + e⁻⁸ˣ⁻ʸ by separation of variables, we need to separate the variables on each side of the equation.

Starting with the left side, using logarithmic identities, we can rewrite it as ln(eˣy) dy = dx.

Next, we integrate both sides with respect to their respective variables. On the left side, we integrate ln(eˣy) dy as (1/x) * eˣy + C1, where C1 is the constant of integration. On the right side, we integrate dx as x + C2, where C2 is another constant of integration.

Finally, setting the two integrals equal to each other, we get (1/x) * eˣy + C1 = x + C2, and we can solve for y.

User SaravInfern
by
8.4k points