Nonlinear ODE

Calculus Level 3

$\begin{cases} \begin{aligned} \dfrac {dy}{dx} & = \dfrac y{x(y+1)} \\ y(1) & = 1 \end{aligned} \end{cases}$

Given the differential equation above, solve for $y$ .

Notation: $W(\cdot)$ denotes the Lambert W-function .

W(ex)

W(e^2x)

W \left(\frac xe\right)

e^2x

1 solution

Chew-Seong Cheong
Dec 17, 2019

$\begin{aligned} \frac {dy}{dx} & = \frac y{x(y+1)} \\ \int \frac {y+1}y \ dy & = \int \frac 1x \ dx \\ y + \ln y & = \ln x + \blue C & \small \blue{\text{where }C = \ln c_1 \text{ is the constant of integration.}} \\ y & = W(c_1x) & \small \blue{\text{where }W(\cdot) \text{ denotes the Lambert W-function.}} \\ y(1) & = 1 & \small \blue{\text{Given}} \\ \implies y(1) & = \ln c_1 = 1 & \small \blue{\text{From }y + \ln y = \ln x + \ln c_1} \\ c_1 & = e \\ \implies y(x) & = \boxed{W(ex)} \end{aligned}$

Nonlinear ODE

1 solution

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