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Calculus

Differential Equations

Order, degree, and solution forms for differential equations including separable and second-order linear cases.

A differential equation relates a function to its derivatives. Order (highest derivative) and degree (power of that derivative) classify the problem. Standard techniques are separation of variables, substitution, and matching characteristic solution forms. For linear equations with constant coefficients, the characteristic equation dictates the form: real distinct roots give a sum of exponentials, repeated roots give exponentials times polynomials, complex roots give sines and cosines (optionally damped by an exponential). The general solution of an n-th-order linear equation is an n-parameter family. Initial or boundary conditions fix the constants.

A differential equation contains derivatives of the unknown function:

 (dydx,d2ydx2) \text{ }\left(\frac{dy}{dx},\frac{d^{2}y}{dx^{2}}\right)\text{ }

or differentials:

 (dx,dy) \text{ }\left(dx,dy\right)\text{ }

is called a differential equation.

The order is the highest derivative present (here, second order):

 (d2ydx2)3+(dydx)=0  is 2\text{ }\left(\frac{d^{2}y}{dx^{2}}\right)^{3}+\left(\frac{dy}{dx}\right)=0\text{ }\text{ }is\text{ }2

The degree is the exponent on the highest-order derivative (the source labels this example 3):

 (dydx)5+y(d2ydx2)=0  is 3\text{ }\left(\frac{dy}{dx}\right)^{5}+y\left(\frac{d^{2}y}{dx^{2}}\right)=0\text{ }\text{ }is\text{ }3

To form a differential equation by eliminating n arbitrary constants, differentiate n times.

A separable first-order equation in the form M(x) dx + N(y) dy = 0:

 fxdx+f2(y)dy=0 \text{ }f_{x}dx+f_{2}\left(y\right)dy=0\text{ }

Integrate the two sides independently:

 ∫f1xdx+∫f2(y)dy=c \text{ }\int f_{1}xdx+\int f_{2}\left(y\right)dy=c\text{ }

For equations not directly separable, a substitution reduces it to separable form:

 x+y=u or y=vx \text{ }x+y=u\text{ }or\text{ }y=vx\text{ }

For a combination of exponentials with opposite-sign exponents:

 u=Aenx+Be−nx \text{ }u=Ae^{nx}+Be^{-nx}\text{ }

satisfies the second-order linear ODE with positive characteristic root n2:

 (d2ydx2)=n2y \text{ }\left(\frac{d^{2}y}{dx^{2}}\right)=n^{2}y\text{ }

A linear combination of two distinct real exponentials:

 y=Aenx+Bemx \text{ }y = Ae^{nx}+Be^{mx}\text{ }

satisfies the second-order ODE with characteristic roots m and n:

 (d2ydx2)−(m+n)(dydx)+mny=0 \text{ }\left(\frac{d^{2}y}{dx^{2}}\right)-\left(m+n\right)\left(\frac{dy}{dx}\right)+mny=0\text{ }

Sinusoidal solutions for complex-conjugate characteristic roots:

(i) y=Asin⁡(nx+B)\left(i\right)\text{ }y=A\sin\left(nx+B\right)

Cosine form, equivalent up to phase:

(ii) y=Acos⁡(nx+B)\left(ii\right)\text{ }y=A\cos\left(nx+B\right)

General harmonic solution:

(iii) y=Asin⁡(nx+B)+Bcos⁡(nx)\left(iii\right)\text{ }y=A\sin\left(nx+B\right)+B\cos\left(nx\right)

All three satisfy the simple harmonic oscillator equation:

(d2ydx2)=−n2y\left(\frac{d^{2}y}{dx^{2}}\right)=-n^{2}y
k > 0: growk > 0: decayk < 0: oscillate
Sign of k in y'' = ky determines whether solutions grow exponentially or oscillate.