The solution of ( dy d x )=2 x -5 y , y (0)=0 is

Step-by-step solution

The differential equation ( dy dx ) = 2x - 5y is equivalent to dy dx = e^ 2x - 5y = e^ 2x e^ -5y . Separating variables yields e^ 5y dy = e^ 2x dx . Integrating both sides gives e^ 5y 5 = e^ 2x 2 + C . Applying the initial condition y(0) = 0 produces 1 5 = 1 2 + C , so C = - 3 10 . Thus, e^ 5y 5 = e^ 2x 2 - 3 10 . Multiplying through by 10 and rearranging terms leads to 5e^ 2x - 2e^ 5y = 3 . This matches option B .