Find the order of the family of curves `y=(c_1+c_2)e^x+c_3e^(x+c_4)`

The differential equation which represents the family of curves y=c_(1)e^(c_(2^(x) where c_(1)andc_(2) are arbitary constants is

The differential equation corresponding to the family of curves y=e^x (ax+ b) is

The orthogonal trajectories of the family of curves y=Cx^(2) , (C is an arbitrary constant), is

The order of the differential equation of family of curves y=C_(1)sin^(-1)x+C_(2)cos^(-1)x+C_(3)tan^(-1)x+C_(4)cot^(-1)x (where C_(1),C_(2),C_(3) and C_(4) are arbitrary constants) is

The value of k such that the family of parabolas y=cx^(2)+k is the orthogonal trajectory of the family of ellipse x^(2)+2y^(2)-y=c, is

Show that the area of the triangle formed by the lines y= m_(1) x + c_(1) , y= m_(2) x + c_(2) and x = 0 is ((c_1 - c_2)^2)/( 2| m_1 - m_2 | ) .

Statement 1 : The order of the differential equation whose general solution is y==c_1cos2x+c_2sin^2x+c_3cos^2x+c_4e^(2x)+c_5e^(2x+c_6) is 3. Statement 2 : Total number of arbitrary parameters in the given general solution in the statement (1) is 3.

Find those values of c for which the equations: 2x+3y=3,(c+ 2)x + (c +4)y=(c+6),(c+2)^2x+(c+4)^2y=(c+6)^2 are consistent.Also solve above equations for these values of c.

Which of the following differential equations has y = c_(1)e^(x) + c_(2)e^(-x) as the general solution?

Find the differential equation whose solution represents the family : c (y + c)^2 = x^3