Differential equation Family of curves orthogonal

shivers20

Junior Member
Joined
Mar 3, 2006
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68
Is this correct?

Find the equation for the family of curves orthogonal to the one-parameter family y=e^cx

ANS:

y= ecx -------------- (1)

Differentiating with respect to x,

dy/dx = cecx ------------ (2)

From (1), ln(y) = cx → c= ln(y)/x

Put c= ln(y)/x and y=ecx in (2) we get,

dy/dx = ln(y)/x * y

dy/dx = y ln(y)/x

Now replace dy/dx by -1/(dy/dx) we get,

-1/(dy/dx) = yln(y)/x

-x = yln(y) dy/dx

-xdx = yln(y)dy

xdx + yln(y)dy = 0

Integrating,

∫xdx + ∫yln(y)dy = K

x2/2 + ln(y)* ∫ydy - ∫d/dy(lny) *∫y dy = K

x2/2 + ln(y)* y2/2 - ∫1/y *y2/2 dy = K

x2/2 + ln(y)* y2/2 – ½ ∫y dy = K

x2/2 + ln(y)* y2/2 – ½ *y2/2 = K

x2/2 +1/2 * y2 ln(y) – 1/4*y2 = K
 
Yes it is, well done! It's not necessary, but one more step to simplify it would be multiplying both sides by 4 and replacing 4K by another arbitrary constant K.
 
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