Question

The differential equation which represents the family of curves y = e^Cx is
(a) y₁ = C² y
(b) xy₁ − ln y = 0
(c) x ln y = yy₁
(d) y ln y = xy₁

Answer 1

(d) y ln y = xy₁

We have,
y = e^Cx
Taking ln on both sides, we get
ln y = Cx ln e
$\Rightarrow \ln y=Cx.....\left(1\right)$
Differentiating both sides of (1) with respect to x, we get
$\frac{1}{y}{y}_1=C$
Substituting the value of C in (1), we get
$$\begin{gathered} \ln y=\frac{y_1}{y}x \\ \Rightarrow y\ln y=y_{1}x \end{gathered}$$