Let y=y(x) be the solution of the differential equation d y d x +2 y ^2 x=2 ^2 x+3 x ^2 x such that y (0)= 5 4 . Then 12 ( y ( 4 )- e ⁻² ) is equal to _______.

Step-by-step solution

aligned I.F. & = e ^ 2 ^2 x d x & = e ^ 2 x aligned Solution of diff. eq. gathered y e^ 2 x = e^ 2 x (2 ^2 x+3 x ^2 x ) d x y e^ 2 x = e^ 2 x (2 ^2 x ) d x+ e^ 2 x (3 x ^2 x ) d x gathered y. e ^ 2 x = e ^ 2 x 2 x- e ^ 2 x 2 ^2 x 2 x d x+ e ^ 2 x 3 x ^2 x d x aligned & y e^ 2 x =2 x e^ 2 x - e ^ 2 x x ^2 x d x & y . e^ 2 x =2 x e^ 2 x - x e^ 2 x 2 + e^ 2 x 4 +C aligned aligned & y=2 x- x 2 + 1 4 + Ce ^ -2 x & x=0, y= 5 4 & c=1 aligned y ( 4 )= 7 4 + e ⁻² Then 12 (y ( 4 )- e ⁻² )=12 ( 7 4 )=21