For all x>0 , let y₁(x), y₂(x) , and y₃(x) be the functions satisfying aligned & d y₁ d x -( x)^2 y₁=0, y₁(1)=5 & d y₂ d x -( x)^2 y₂=0, y₂(1)= 1 3 & d y₃ d x - ( 2-x^3 x^3 ) y₃=0, y₃(1)= 3 5 e aligned respectively. Then _ x 0⁺ y₁(x) y₂(x) y₃(x)+2 x e^ 3 x x is equal to _____

Step-by-step solution

aligned & d y₁ y₁ + d y₂ y₂ + d y₃ y₃ = ( ^2 x+ ^2 x+ 2-x^3 x^3 ) d x & (y₁ y₂ y₃ )= -1 x^2 +C & (y₁(x) y₂(x) y₃(x) )= -1 x^2 +C & (5 1 3 3 5 e )= -1 x^2 +C & C=0 aligned aligned & y₁(x) y₂(x) y₃(x)=e^ -1 x^2 & _ x 0 e^ -1 x^2 +2 x e^ 3 x x & _ x 0 1 e^ 3 x+ 1 x^2 x + _ x 0 2 x e^ 3 x x aligned _ x 0 1 e^ 3 x x x e^ 1 x^2 1 x +2=0+2 _ y e ^ y ^2 y = e ^ y ^2 2 y 1 = =2