Question

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b:y= e^2x (a+bx)

Answer 1

y= e^2x (a+bx)....(1)
Differentiating both sides with respect to x, we get:
y'= 2e^2x (a+bx)+e^2x .b

⇒y'= e^2x (2a+2bx+b)...(2)

Multiplying equation(1) with equation (2) and then subtracting it from equation (2),we get:

y'-2y=e^2x (2a+2bx+b)-e^2x (2a+2bx)

\(\Rightarrow\) y'-2=be^2x...(3)

Differentiating both sides with respect to x, we get:

y''k-2y'=2be^2x....(4)

Dividing equation(4)by equation(3),we get:

\(\frac{y''-2y''}{y'-2y}=2\)

\(\Rightarrow\) y''-2y'=2y'-4y

\(\Rightarrow\) y''-4y'+4y=0

This is the required differential equation of the given curve.