Order, Degree, Formation, and Verification

The vocabulary you must have cold:

• Differential equation: an equation involving derivatives of an unknown function, e.g. $\dfrac{dy}{dx} = 3x$ or $\dfrac{d^2y}{dx^2} + 4y = 0$.
• Order: the order of the highest derivative present.
• Degree: the power of the highest-order derivative once the equation is polynomial in its derivatives.
• Arbitrary constants: free parameters ($a, b, c, C_1, \dots$) in a solution family.
• General solution: contains as many independent arbitrary constants as the order.
• Particular solution: a general solution with its constants fixed by given conditions.

Order of a differential equation $=$ the order of the highest-order derivative that appears in it.

• $\dfrac{d^2y}{dx^2}$ present but no higher derivative $\Rightarrow$ order 2, regardless of any power on it.
• A high power on a LOW derivative does not raise the order: $\big(\tfrac{dy}{dx}\big)^{5} + \tfrac{d^3y}{dx^3} = 0$ is order 3 (because $\tfrac{d^3y}{dx^3}$ is present), not order 5.
• Mixed powers of the same top derivative also leave the order alone.

To find the degree: 1. Clear all radicals and fractional powers on the derivatives (raise to a suitable power). 2. Once the equation is polynomial in the derivatives, the degree is the power on the highest-order derivative.

• Example shape: $\sqrt{y''} = \sqrt[5]{y' - 5}$ becomes $(y'')^5 = (y'-5)^2$ after raising to the 10th power $\Rightarrow$ degree 5.
• The LCM of the fractional exponents tells you the power to raise both sides to.

Degree is undefined when the equation cannot be made polynomial in its derivatives:

• A derivative appears inside a transcendental function: $\log\!\big(\tfrac{d^2y}{dx^2}\big)$, $\sin\!\big(\tfrac{dy}{dx}\big)$, $e^{\,y''}$, etc.
• Order is still well-defined in these cases — read it as usual (the highest derivative present).
• Only radicals/fractional powers can be cleared; a derivative inside $\log$/$\sin$/$\cos$/$e^{(\cdot)}$ is permanent.

Constants merge in predictable ways — spot and collapse them:

• Sums merge: $C_1 + C_2 \to A$ (one constant), and $C_1 + C_3 \to A$.
• Exponential shifts absorb: $C_3 e^{x + C_4} = (C_3 e^{C_4})e^x = B e^x$ — the $C_4$ vanishes into a single $B$.
• Same-form terms merge: $(C_1 + C_2)e^x = A e^x$; two constants become one.
• After collapsing, order = number of surviving independent constants.

The formation recipe:

• Count the independent arbitrary constants $n$ in the family (collapse redundant ones first).
• **Differentiate the family $n$ times, then eliminate** all $n$ constants using the original equation plus the derived equations.
• The result is a differential equation of **order $n$**, free of arbitrary constants.
• The degree of that equation is read afterwards (clear radicals first).

For a family with constants, differentiate as many times as there are constants, then eliminate:

• One constant: differentiate once, solve for the constant, substitute back.
• Two constants (e.g. $Ax^2 + By^2 = 1$): differentiate twice and eliminate $A, B$, giving a second-order equation.
• Keep known functions: in $x^2 y = 4e^x + c$, the $e^x$ is a known function, NOT the arbitrary constant — only $c$ is eliminated, so $e^x$ survives in the answer.
• For $y = e^x(a + bx + x^2)$: use $y = e^x u$, differentiate, and eliminate $a, b$.

Set up the standard form from the geometric description, then eliminate:

• Circles, centre on X-axis, through origin: $(x-a)^2 + y^2 = a^2 \Rightarrow x^2 + y^2 = 2ax$; eliminate $a$ $\to$ $y^2 = x^2 + 2xy\,y'$ (order 1, one constant).
• Circles through origin, centre on Y-axis: $x^2 + y^2 = 2ky \Rightarrow (x^2 - y^2)y' - 2xy = 0$.
• Circles touching Y-axis at origin, centre on X-axis: $x^2 + y^2 = 2hx \Rightarrow x^2 - y^2 + 2xy\,y' = 0$.
• Parabolas, vertex origin, axis along +Y: $x^2 = 4ay \Rightarrow x\dfrac{dy}{dx} = 2y$ (one constant $a$, order 1).
• All parabolas, axis parallel to Y: three constants $\Rightarrow$ order 3, $\dfrac{d^3y}{dx^3} = 0$.

Three verification tasks, all by substitution:

• Confirm a solution: compute $y', y''$ from the given $y$, plug into the ODE, and check the equation holds identically.
• **Find a constant $k$:** for a PARAMETRIC solution $x = x(t),\ y = y(t)$, use $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$ and $\dfrac{d^2y}{dx^2} = \dfrac{d}{dx}\!\big(\tfrac{dy}{dx}\big)$ to substitute, then solve for $k$.
• Identify the family: solve/simplify the given ODE to its solution curve and name it (circle, hyperbola, ellipse, pair of lines).