Differential Equations Spellbook

Welcome, young math wizards! ✨ Today we'll master the magical art of differential equations.

Each equation is like a spell - we must determine its OrderThe highest derivative present in the equation and DegreeThe power of the highest order derivative when expressed as a polynomial to understand its power!

Click on any spell to reveal its secrets!

(i) \(\frac{dy}{dx} + xy = \cot x\)
This spell contains first-order magic (\(\frac{dy}{dx}\))
The power of \(\frac{dy}{dx}\) is clearly 1
No other derivatives are present
Order: 1 (First Order Spell)
Degree: 1 (Linear Magic)
(ii) \(\left( \frac{d^3 y}{dx^3} \right)^{\frac{2}{3}} - 3 \frac{d^2 y}{dx^2} + 5 \frac{dy}{dx} + 4 = 0\)
Powerful third-order magic (\(\frac{d^3 y}{dx^3}\)) detected!
To determine degree, we must eliminate fractional exponents
Raise both sides to the 3rd power: \(\left( \frac{d^3 y}{dx^3} \right)^2 - \text{(other terms)} = 0\)
Now the highest derivative has power 2
Order: 3 (Third Order Spell)
Degree: 2 (Quadratic Magic)
(iii) \(\left( \frac{d^2 y}{dx^2} \right)^2 + \left( \frac{dy}{dx} \right)^2 = x \sin \left( \frac{d^2 y}{dx^2} \right)\)
Second-order magic (\(\frac{d^2 y}{dx^2}\)) is present
The sine function makes this non-polynomial
Degree is undefined for non-polynomial spells
Order: 2 (Second Order Spell)
Degree: Undefined (Mystical Magic)
(iv) \(\sqrt{\frac{dy}{dx}} - 4 \frac{dy}{dx} - 7x = 0\)
First-order magic (\(\frac{dy}{dx}\)) detected
Square both sides to remove square root: \(\frac{dy}{dx} = (4 \frac{dy}{dx} + 7x)^2\)
Expanding gives polynomial in derivatives
Highest power of \(\frac{dy}{dx}\) is 2
Order: 1 (First Order Spell)
Degree: 2 (Quadratic Magic)
(v) \(y \left( \frac{dy}{dx} \right) = \frac{x}{\left( \frac{dy}{dx} \right) + \left( \frac{dy}{dx} \right)^3}\)
First-order magic (\(\frac{dy}{dx}\)) throughout
Multiply both sides by denominator: \(y \left( \frac{dy}{dx} \right)^4 + y \left( \frac{dy}{dx} \right)^2 = x\)
Highest power of \(\frac{dy}{dx}\) is 4
Order: 1 (First Order Spell)
Degree: 4 (Quartic Magic)
(vi) \(x^2 \frac{d^2 y}{dx^2} + \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^{\frac{1}{2}} = 0\)
Second-order magic (\(\frac{d^2 y}{dx^2}\)) appears
Square root makes equation non-polynomial
Degree is undefined for non-polynomial spells
Order: 2 (Second Order Spell)
Degree: Undefined (Mystical Magic)
(vii) \(\left( \frac{d^2 y}{dx^2} \right)^3 = \sqrt{1 + \left( \frac{dy}{dx} \right)}\)
Second-order magic (\(\frac{d^2 y}{dx^2}\)) is strongest here
Square both sides to eliminate square root
New equation: \(\left( \frac{d^2 y}{dx^2} \right)^6 = 1 + \frac{dy}{dx}\)
Highest power of \(\frac{d^2 y}{dx^2}\) is 6
Order: 2 (Second Order Spell)
Degree: 6 (Hexic Magic)
(viii) \(\frac{d^2 y}{dx^2} = xy + \cos \left( \frac{dy}{dx} \right)\)
Second-order magic (\(\frac{d^2 y}{dx^2}\)) detected
Cosine function makes this non-polynomial
Degree is undefined for non-polynomial spells
Order: 2 (Second Order Spell)
Degree: Undefined (Mystical Magic)
(ix) \(\frac{d^2 y}{dx^2} + 5 \frac{dy}{dx} + \int y dx = x^3\)
Second-order magic (\(\frac{d^2 y}{dx^2}\)) is present
Integral makes this an integro-differential equation
Degree is not defined for equations with integrals
Order: 2 (Second Order Spell)
Degree: Undefined (Mystical Magic)
(x) \(x = e^{xy \left( \frac{dy}{dx} \right)}\)
First-order magic (\(\frac{dy}{dx}\)) in exponential
Take natural log: \(xy \frac{dy}{dx} = \ln x\)
Highest power of \(\frac{dy}{dx}\) is 1
Order: 1 (First Order Spell)
Degree: 1 (Linear Magic)