Quick questions on Differential equations explained: H2 Mathematics Calculus

If $\dfrac{dy}{dx} = \mathrm{f}(x)$ (the right side depends only on $x$), integrate directly:

If $\dfrac{dy}{dx} = \mathrm{g}(x)\mathrm{h}(y)$, separate the variables to opposite sides and integrate:

When separation produces a rational function of $y$ on the left, partial fractions often make it integrable. The logistic model $\dfrac{dP}{dt} = kP(1 - P)$ separates to $\displaystyle\int \dfrac{1}{P(1 - P)}\,dP = \int k\,dt$, and the left integrand splits as $\dfrac{1}{P} + \dfrac{1}{1 - P}$, each a logarithm. This links the differential-equations work directly to the partial-fractions and integration techniques: a separable equation is only as solvable as the integrals it produces, so recognising when partial fractions are needed is part of the method.

The general solution of a first-order equation is a family of curves, one for each value of the constant, and a boundary condition selects the single curve through a given point. Sketching a few members of the family, then highlighting the particular solution, is a common H2 task that checks understanding rather than algebra. For $\dfrac{dy}{dx} = ky$, the family is a set of exponential curves $y = Ae^{kx}$ of different heights, and the condition fixes which one passes through the stated point. Reading the constant as "which curve in the family" makes the general-versus-particular distinction concrete.

Integrating both sides needs only a single combined constant $C$.

Solve $\dfrac{dy}{dx} = 3x^2$ with $y = 1$ at $x = 0$. [2 marks]

Solve $\dfrac{dy}{dx} = ky$ in general form. [2 marks]

State the difference between a general and a particular solution. [2 marks]