Let y : (- , ) (0, ) be the solution of the differential equation dy dx = e^ 5x y^3 + y^3 e^x + e^x y^4 , satisfying y(0) = 1 2 . Then the value of y( _e 2) is

Step-by-step solution

The given differential equation is dy dx = e^ 5x y^3 + y^3 e^x + e^x y^4 Factoring the numerator and denominator, we get dy dx = y^3 (e^ 5x + 1) e^x (1 + y^4) Separating the variables x and y , we obtain 1 + y^4 y^3 dy = e^ 5x + 1 e^x dx ( 1 y^3 + y ) dy = (e^ 4x + e^ -x ) dx Integrating both sides yields ( y⁻³ + y ) dy = (e^ 4x + e^ -x ) dx - 1 2y^2 + y^2 2 = e^ 4x 4 - e^ -x + C Multiplying by 2 , we get y^2 - 1 y^2 = e^ 4x 2 - 2e^ -x + 2C Using the initial condition y(0) = 1 2 , we substitute x = 0 and y = 1 2 :