Question

Solve the initial-value problem

$\frac{dy}{dx}=2y\left(1-5y\right),y\left(0\right)=1$

Short answer

The solution is$y\left(x\right)=\frac{e^{2x}}{5e^{2x}-4}$

Step-by-step solution

Step 1. Given information 

given value is$\frac{dy}{dx}=2y\left(1-5y\right)$

Step 2. Solve equation

$$\begin{gathered} \int \frac{dy}{y\left(1-5y\right)}=2\int dx \\ \frac{1}{y\left(1-5y\right)}=\frac{A}{y}+\frac{B}{1-5y} \\ A=1 \\ B=5 \\ \frac{1}{y\left(1-5y\right)}=\frac{1}{y}+\frac{5}{1-5y} \\ \int \left[\frac{1}{y}+\frac{5}{1-5y}\right]dy=2x+c \\ \ln \left(y\right)-\ln \left(\left|1-5y\right|\right)=2x+c \\ \ln \left(\left|\frac{y}{1-5y}\right|\right)=2x+c \\ \frac{y}{1-5y}=e^{2x+c} \\ \frac{1-5y}{y}=e^{-2x-c} \\ =A{e}^{-2x} \\ \frac{1}{y}=5+A{e}^{-2x} \\ y=\frac{1}{5+A{e}^{-2x}} \\ 1=\frac{1}{5+A} \\ 5+A=1 \\ A=-4 \\ y\left(x\right)=\frac{1}{5-4e^{-2x}} \\ y\left(x\right)=\frac{e^{2x}}{5e^{2x}-4} \end{gathered}$$

Step 3. The solution

The solution is$y\left(x\right)=\frac{e^{2x}}{5e^{2x}-4}$