DE 1003: solving DE by a given substitution

In this question, we solve a DE using a given substitution.

Question types

1. Obtaining $\displaystyle \frac{\mathrm{d}z}{\mathrm{d}x} = f(x)$ using the substitution $z=h(x)g(y)$
2. Obtaining $\displaystyle \frac{\mathrm{d}z}{\mathrm{d}x} = f(z)$ using the substitution $z=h(x)g(y)$
3. Obtaining $\displaystyle \frac{\mathrm{d}z}{\mathrm{d}x} = f(z)$ using the substitution $z=ax+by$
4. Solving $\displaystyle \frac{\mathrm{d}^2y}{\mathrm{d}x^2} + a\frac{\mathrm{d}y}{\mathrm{d}x}+b = 0$ using the substitution $z=\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}$
5. Solving $\displaystyle x\frac{\mathrm{d}y}{\mathrm{d}x} = my-b$ using the substitution $y=zx^m$

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