Ordinary Differential Equations (ODEs)

A first-order ODE is called separable if it can be written as $y' = f(x)g(y)$. The solution method consists of separating variables and integrating:

$$\int\frac{dy}{g(y)} = \int f(x),dx \qquad (\text{assuming }g(y)\neq0).$$

This gives an implicit relation between $y$ and $x$, which can sometimes be solved explicitly for $y$.

Exponential growth/decay: $y' = ky$ ($k$ constant). Separating: $dy/y = k,dx$, integrating: $\ln|y| = kx + c$, exponentiating: $y = y_0 e^{kx}$. $k>0$: growth (population, compound interest); $k<0$: decay (radioactivity, cooling).

Newton's law of cooling: the rate of change of temperature is proportional to the difference from ambient: $T' = -k(T-T_{\text{amb}})$. Setting $u=T-T_{\text{amb}}$ gives $u'=-ku$, hence $u=u_0e^{-kt}$, i.e., $T = T_{\text{amb}} + (T_0-T_{\text{amb}})e^{-kt}$.

Cauchy (initial value) problem: an ODE together with an initial condition $y(x_0)=y_0$. The initial condition selects a unique solution from the general family by fixing the constant $C$.

The homogeneous equation $ay''+by'+cy=0$ with constant coefficients $a,b,c\in\mathbb{R}$ is solved by seeking solutions of the form $y=e^{\lambda x}$. Substituting gives the characteristic equation $a\lambda^2+b\lambda+c=0$.

The discriminant $\Delta = b^2-4ac$ determines three cases:

• $\Delta>0$ (two distinct real roots $\lambda_1,\lambda_2$): solution $y=C_1 e^{\lambda_1 x} + C_2 e^{\lambda_2 x}$. Two linearly independent exponentials.
• $\Delta=0$ (repeated root $\lambda=-b/(2a)$): solution $y=(C_1+C_2 x)e^{\lambda x}$. The factor $x$ provides the second independent solution.
• $\Delta<0$ (complex conjugate roots $\lambda=\alpha\pm i\beta$): solution $y=e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)$. The solution oscillates with exponentially modulated amplitude — the basis for modelling harmonic oscillators.

For the non-homogeneous equation $ay''+by'+cy=f(x)$, the general solution is $y=y_h+y_p$ where $y_h$ solves the homogeneous equation and $y_p$ is a particular solution (found by undetermined coefficients or variation of parameters).