The equation of the curve passing through (1,2) and whose tangent at any point (x, y) makes an angle ⁻¹(2 x+3 y) with the X -axis is .........

Step-by-step solution

P =(1,2) aligned Inclination ( ) & = ⁻¹(2 x+3 y) & =2 x+3 y Slope (m) & = aligned gathered d y d x =2 x+3 y d y d x -3 y=2 x P=-3 and Q=2 x IF =e^ P d x =e^ -3 d x =e^ -3 x gathered General soution is, aligned y( IF ) & = Q( IF ) d x+c y (e^ -3 x ) & = 2 x e^ -3 x d x+c y (e^ -3 x ) & =2 x e^ -3 x d x- ( d d x (2 x) e^ -3 x d x ) d x+c & =2 x e^ -3 x -3 - 2 e^ -3 x -3 d x+c & = -2 3 x e^ -3 x + 2 3 e^ -3 x d x+c & = -2 3 x e^ -3 x + 2 3 e^ -3 x -3 +c y e^ -3 x & = -2 3 e^ -3 x (x+ 1 3 )+c aligned Multiply by e^ 3 x