Given: \(\displaystyle{y}^{{2}}={c}{e}^{{x}}-{x}-{1}\) Find the orthogonal family of curves



Answered question


Given: y2=cexx1
Find the orthogonal family of curves and write 2 curves from both families that pass in (2,0).

Answer & Explanation

Makenzie Hart

Makenzie Hart

Beginner2022-03-25Added 8 answers

Step 1
To find the orthogonal family of curves, you need to first solve for c in terms of x and y. 
From the given family of curves, c=1+x+y2ex
So, y=x+y22y
Slope for the orthogonal trajectories will be y=2yx+y2
Or, 2y dx +(x+y2) dy =0, which is of the form M dx +N dy =0.
Step 2
To solve the above differential equation, note that it is not exact (MyNx) so first find an integrating factor that makes it exact, which in this case is 1y.
Multiplying by the integrating factor,
2y dx +(xy+y32) dy =0
As My=Nx, it is now exact and integrating, we get the solution
To get two curves from both families that pass through the point (2,0), plug in x=2,y=0 in the equation of curves and find values of c and C.

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