Please help me with a differential equations question

Question

Find the orthogonal trajectories of the family of curves 𝑦 = 𝑐𝑒 ^x .

please write the solution with full steps.

Answer 1

Given equation is y =ce^x

Now find value of c in terms of x and y i.e. c= y/(e^x)

Now find the slope of the given family, i.e. dy/dx = ce^x

Now substitute for c, then we get, dy/dx = y

So, the slope for the given family of equations is y

Now, we know that if two families are orthogonal then the product of their slopes is -1.

Thus the slope for the orthogonal family will be dy/dx =-1/y

Now integrating to get equation for the orthogonal family,

We get the orthogonal family to be y^2/2 = -x^2/2 +c

i.e. the orthogonal family is x^2 + y^2 = 2c