Question

Hii


Find the differential equation of the family of straight lines

y=mx+ a/m

where a is the parameter.

Answer 1

with parametet m

If we have an equation f(x,y,c1,c2,....cn)=uf(x,y,c1,c2,....cn)=uContaining n arbitrary constant c1,c2...cnc1,c2...cn, then by differentiating n times, we get (n+1)(n+1) equations in total. If we eliminate the arbitrary constants c1,c2....cn,c1,c2....cn, we get a D.E of order n

Step 1:
y=mx+amy=mx+am where m is the parameter
dydxdydx=m=m -----(ii)

Step 2:
Substitute for m from (ii) in (i)
∴y=xdydx+adydx∴y=xdydx+adydx
(ie) x(dydx)2x(dydx)2−ydydx−ydydx+a=0+a=0
D. E is required.

hope it helps you dear..