Advanced Non-Linear First-Order Equations & Singular Solutions

1. Standard Form of Bernoulli's Equation

An equation of the form:

where $P(x)$ and $Q(x)$ are continuous functions and $n \in \mathbb{R}$ is any real constant, is called Bernoulli's equation. If $n = 0$, it is linear non-homogeneous. If $n = 1$, it is linear homogeneous and separable. For any $n \neq 0, 1$, it is strictly non-linear.

2. The Power Transformation Method

Dividing the equation by $y^n$:

Introduce the change of variable $v = y^{1-n}$. Differentiating with respect to $x$ using the chain rule:

Substituting into the equation transforms it into an exact first-order linear ODE in $v$:

This linear equation is solved using integrating factor $\mu(x) = \exp\left((1 - n) \int P(x) dx\right)$, followed by back-substitution $v = y^{1-n}$.

1. The General Riccati Equation

A non-linear differential equation of the quadratic form:

is called a Riccati equation. In general, Riccati equations cannot be solved by elementary quadratures unless at least one particular solution is already known.

2. Reduction Given One Known Particular Solution $y_1(x)$

If a particular solution $y_1(x)$ is known (so that $y_1' = P y_1^2 + Q y_1 + R$), the general solution is obtained by the substitution:

Differentiating and substituting into Riccati's equation:

Subtracting $y_1'$ and multiplying by $-v^2$ reduces the equation to a standard linear first-order ODE in $v$:

3. Transformation to a Second-Order Linear Equation

Making the substitution $y = -\frac{1}{P(x) u} \frac{du}{dx}$ transforms the non-linear first-order Riccati equation into a homogeneous second-order linear ODE in $u(x)$:

1. First-Order Equations of Higher Degree ($F(x, y, p) = 0$, $p = dy/dx$)

When the first derivative appears to powers greater than 1, we write $p = \frac{dy}{dx}$. Three primary classical methods solve such equations:

1. Equations Solvable for $p$: If $F(x, y, p) = 0$ can be factored as $[p - f_1(x, y)][p - f_2(x, y)] \dots [p - f_k(x, y)] = 0$, then each factor $p = f_i(x, y)$ is solved independently to yield solutions $\phi_i(x, y, C) = 0$. The composite general solution is $\prod_{i=1}^k \phi_i(x, y, C) = 0$.
2. Equations Solvable for $y$: Written as $y = f(x, p)$. Differentiating both sides with respect to $x$ yields $p = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial p} \frac{dp}{dx}$, which is a first-order differential equation in $p$ and $x$.
3. Equations Solvable for $x$: Written as $x = g(y, p)$. Differentiating both sides with respect to $y$ using $\frac{dx}{dy} = \frac{1}{p}$ gives a first-order ODE in $p$ and $y$.

2. Clairaut's Equation

A classic and prominent equation solvable for $y$ is Clairaut's equation:

Differentiating both sides with respect to $x$:

This product produces two distinct solution branches:

• Branch 1 (General Solution): $\frac{dp}{dx} = 0 \implies p = C$ (constant). Substituting $p = C$ into the original equation yields the general solution: $$y = C x + f(C)$$ which represents a one-parameter family of straight lines!
• Branch 2 (Singular Solution / Envelope): $x + f'(p) = 0 \implies x = -f'(p)$. Substituting this into $y = xp + f(p)$ gives the parametric equations of the singular solution: $$x = -f'(p), \quad y = -p f'(p) + f(p)$$ Eliminating $p$ between these relations yields an envelope curve that is tangent to every member of the general solution family of straight lines, yet cannot be obtained from $y = Cx + f(C)$ for any constant $C$.