Direct Integrations [Sec/ IP Level]
•	$\frac{\mathrm{d}y}{\mathrm{d}x}=\mathrm{f}(x)\Rightarrow y=\int{\mathrm{f}(x)}\,\mathrm{d}x$ 
•	$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=\mathrm{f}(x)\Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x}=\int{\mathrm{f}(x)}\,\mathrm{d}x\Rightarrow y=\int{\int{\mathrm{f}(x)}\,\mathrm{d}x\,\mathrm{d}x}$
Method of Separable Variables
•	$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{f}(x)}{\mathrm{g}(y)}\Rightarrow \int{\mathrm{g}(y)\,\mathrm{d}y}=\int{\mathrm{f}(x)\,\mathrm{d}x}$ 
•	$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{g}(y)}{\mathrm{f}(x)}\Rightarrow \int{\frac{1}{\mathrm{g}(y)}\,\mathrm{d}y}=\int{\frac{1}{\mathrm{f}(x)}\,\mathrm{d}x}$
•	$\frac{\mathrm{d}y}{\mathrm{d}x}=\mathrm{g}(y)\mathrm{f}(x)\Rightarrow \int{\frac{1}{\mathrm{g}(y)}\,\mathrm{d}y}=\int{\mathrm{f}(x)\,\mathrm{d}x}$

•	$\frac{\mathrm{d}y}{\mathrm{d}x}=\mathrm{f}(x)\Rightarrow y=\displaystyle\displaystyle \int{\mathrm{f}(x)}\,\mathrm{d}x$ 

•	$\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=\mathrm{f}(x)\Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x}=\displaystyle\displaystyle \int{\mathrm{f}(x)}\,\mathrm{d}x\Rightarrow y=\displaystyle\displaystyle \int{\displaystyle\displaystyle \int{\mathrm{f}(x)}\,\mathrm{d}x\,\mathrm{d}x}$

•	$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{f}(x)}{\mathrm{g}(y)}\Rightarrow \displaystyle\displaystyle \int{\mathrm{g}(y)\,\mathrm{d}y}=\displaystyle\displaystyle \int{\mathrm{f}(x)\,\mathrm{d}x}$ 

•	$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{g}(y)}{\mathrm{f}(x)}\Rightarrow \displaystyle\displaystyle \int{\frac{1}{\mathrm{g}(y)}\,\mathrm{d}y}=\displaystyle\displaystyle \int{\frac{1}{\mathrm{f}(x)}\,\mathrm{d}x}$

•	$\frac{\mathrm{d}y}{\mathrm{d}x}=\mathrm{g}(y)\mathrm{f}(x)\Rightarrow \displaystyle\displaystyle \int{\frac{1}{\mathrm{g}(y)}\,\mathrm{d}y}=\displaystyle\displaystyle \int{\mathrm{f}(x)\,\mathrm{d}x}$



Differential Equations

H2 Math

International Baccalaureate Higher Level Mathematics

Differential Equations

Question abb61e6f

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