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NSW Y12 Maths - Extension 1 Differential Equations Solving y'=f(x)g(y)

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Solving y'=f(x)g(y) Theory

When a differential equation is of the form \(\dfrac{d y}{d x}=f(x) g(y)\) then the variables must be separated so that it is possible to integrate. \\ A simple example is \(\dfrac{d y}{d x}=x^2 y\)\\ This is integrated in the following manner:\\ $\begin{aligned} \int \frac{1}{y} d y & =\int x^2 d x \\ \log _{e }y & =\frac{1}{3} x^3+c \\ y & =e^{\frac{1}{3} x^3+c} \\ \text{OR } \quad y & =A e^{\frac{1}{3} x^3} \quad\left(A=e^c\right) \end{aligned}$\\ \textbf{Example}\\ %question 118337 Find the particular solution to \(\dfrac{dy}{dx}=\dfrac{y}{x}\) at \((1,1)\)\\ \textbf{Solution}\\ $\begin{aligned} \frac{d y}{d x} &=\frac{y}{x} \\ \int \frac{1}{y} d y &=\int \frac{1}{x} d x \\ \ln |y| &=\ln |x|+c \\ \text{At } (1,1)\quad & \ln 1=\ln 1+C \\ \therefore c &=0\\ \therefore \quad \ln |y| &=\ln |x| \\ \therefore|y| &=|x| \\ \therefore y &=\pm x \end{aligned}$\\

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  • Solving y=f(x)g(y) - Video - Separable differential equations

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