Q. 20 - Calculus

Question

$\frac{dy}{dx}=3-4y, y\left(0\right)=2$

Step-by-Step Solution

Answer

The solution is $y\left(x\right)=\frac{1}{4}\left[3+5e^{-4x}\right]$

Step 1. Given information

given differential equation is $\frac{dy}{dx}=3-4y$

Step 2. solve equation

$$\begin{gathered} \int \frac{1}{3-4y}dy=\int dx \\ -\frac{1}{4}\ln \left(\left|3-4y\right|\right)=x+c1 \\ \ln \left(\left|3-4y\right|\right)=-4x-4c1 \\ 3-4y=A{e}^{-4x} \\ y=\frac{1}{4}\left(3-A{e}^{-4x}\right) \\ 2=\frac{1}{4}\left(3-A\right) \\ 8-3=-A \\ A=-5 \\ y\left(x\right)=\frac{1}{4}\left(3+5e^{-4x}\right) \end{gathered}$$

Step 3. The solution

The Solution is $y\left(x\right)=\frac{1}{4}\left[3+5e^{-4x}\right]$