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

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Solving y'=g(y) Theory

When a differential equation is of the form \(y^{\prime}=f(y)\) it is important to rewrite \(y^{\prime}\) in the form \(\dfrac{d y}{d x}\).\\ A simple example is \(y^{\prime}=y\),\\ \( \therefore \dfrac{d y}{d x}=y\)\\ This is then integrated in the following manner $\begin{aligned} \int \dfrac{1}{y} d y&=\int 1\, d x\\ \log _e y&=x+c\\ y&=e^{x+c} \text{, is the general solution.} \end{aligned}$\\ \textbf{Example}\\ %question 118331 Solve \(y'=4y^2\) with \(y(0)=2\)\\ \textbf{Solution}\\ $\begin{aligned} &\frac{d y}{d x}=4 y^2 \\ &\frac{d x}{d y}=\frac{1}{4 y^2} \\ &x=\frac{1}{4} \int y^{-2} d y \\ &x=\frac{1}{4} \times \frac{y^{-1}}{-1}+c \\ &x=-\frac{1}{4 y}+c \end{aligned}$\\ $\begin{aligned} \text { At }x &=0, y=2 \\ 0 &=-\frac{1}{8}+c \rightarrow c =\frac{1}{8}\\ x &=-\frac{1}{4 y}+\frac{1}{8} \\ \frac{1}{4 y} &=\frac{1}{8}-x \\ \frac{1}{4 y} &=\frac{1-8 x}{8} \\ \frac{1}{y} &=\frac{1-8 x}{2} \\ \therefore y &=\frac{2}{1-8 x} \end{aligned}$\\

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  • Solving y=g(y) - Video - Solving differential equations given in terms of f(y)

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NSW Year 12 Maths Extension 1 - Trig Equations

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NSW Year 12 Maths Extension 1 - Proof

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NSW Year 12 Maths Extension 1 -Vectors

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NSW Year 12 Maths Extension 1 - Calculus

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NSW Year 12 Maths Extension 1 -Differential Equations

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NSW Year 12 Maths Extension 1 -Projectile Motion

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NSW Year 12 Maths Extension 1 -Binomial Distribution

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