State True or False for the following:

(i) The integrating factor of a differential equation of the form $\frac{dy}{dx}+P_{1}x=Q_1$ is given by $e^{\int P_{1}dy}.$

(ii) solution of a differential equation of the type $\frac{dy}{dx}+P_{1}x=Q_1$ is given by $y\cdot I.F.=\left(I.F.\right)Q_{1}dy.$

(iii) Correct substitution for solving the differential equation of the type $\frac{dy}{dx}=f\left(x,y\right),$ where $f\left(x,y\right)$ is a homogeneous function of zero degree, is $y=vx$ .

(iv) Correct substitution for the solution of the differential equation of the type dx/dy = g(x, y) where g (x, y) is a homogeneous function of the degree zero is x = vy.

(v) Number of arbitrary constants in the particular solution of a differential equation of order two is two.

(vi) The differential equation representing the family of circles x² + (y – a)² = a² will be of order two.

(vii) The solution of dy/dx = (y/x)^1/3 is y ^2/3-x^2/3 = c.

(viii) Differential equation representing the family of curves

 y e^x (A cos x + B sin x is d² y/dx² – 2dy/dx + 2y = 0

(ix) The solution of the differential equation dy/dx = x + 2y/x is x + y = kx².

 (x) Solution of xdy/dx = y +x tan y/x is sin (y/x) =cx        

(xi) The differential equation of all non- horizontal lines in a plane is d² y/dx²=0