For all x>0, let y_1(x), y_2(x), and y_3(x) be the functions satisfying aligned & d y_1 d x -( x)^2 y_1=0, y_1(1)=5 \\ & d y_2 d x -( x)^2 y_2=0, y_2(1)= 1 3 \\ & d y_3 d x - ( 2-x^3 x^3 ) y_3=0, y_3(1)= 3 5 e aligned respectively. Then _ x 0^ + y_1(x) y_2(x) y_3(x)+2 x e^ 3 x x is equal to _____
A. A
aligned & d y_1 y_1 + d y_2 y_2 + d y_3 y_3 = ( ^2 x+ ^2 x+ 2-x^3 x^3 ) d x \\ & (y_1 y_2 y_3 )= -1 x^2 +C \\ & (y_1(x) y_2(x) y_3(x) )= -1 x^2 +C \\ & (5 1 3 3 5 e )= -1 x^2 +C \\ & C=0 aligned aligned & y_1(x) y_2(x) y_3(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
Related: Mathematics — Differential Equations · All PYQ Banks