[Verse 1]
When functions change and time moves on
Derivatives tell us what's going wrong
Ordinary equations, first order start
Existence and uniqueness, that's the art
Linear systems dancing in their phase
Portrait shows the flow in every case
[Chorus]
Differential equations, how things change with time
Partial derivatives in space and rhyme
From heat to waves to chaos in between
Mathematical stories yet unseen
Solve the mystery, find the flow
Where the functions always go
[Verse 2]
Partial derivatives take the stage
Wave and heat across the page
Laplace whispers through the field
Separation variables revealed
Classify them one by one
Hyperbolic, parabolic, done
[Chorus]
Differential equations, how things change with time
Partial derivatives in space and rhyme
From heat to waves to chaos in between
Mathematical stories yet unseen
Solve the mystery, find the flow
Where the functions always go
[Verse 3]
Sobolev spaces hold the weak
Solutions that the strong can't reach
Elliptic stable, parabolic smooth
Hyperbolic waves on the move
Theory built on solid ground
Where mathematical truth is found
[Bridge]
Dynamical systems spin around
Stability can be found
Bifurcation changes course
Chaos brings a strange new force
Attractors pull with Lyapunov might
Exponential growth takes flight
[Verse 4]
Calculus of variations seeks the best
Euler-Lagrange puts it to the test
Direct methods find the way
Optimal control saves the day
Action principle guides the path
Through the realm of higher math
[Chorus]
Differential equations, how things change with time
Partial derivatives in space and rhyme
From heat to waves to chaos in between
Mathematical stories yet unseen
Solve the mystery, find the flow
Where the functions always go
[Verse 5]
Fluid dynamics flows so free
Navier-Stokes complexity
Euler equations guide the stream
Turbulence fulfills the dream
Millennium problems yet unsolved
In these equations all involved
[Outro]
Einstein's field equations bend
Space and time that never end
Singularities hide away
Penrose diagrams light the way
From the small to cosmic scale
Differential truth will never fail