furthermathskit 1.0.0

Step-by-step solver for Further Mathematical Methods — ODEs, PDEs, Fourier series, linear algebra, with LaTeX output in Jupyter

furthermathskit

Step-by-step solver for Further Mathematical Methods

Renders full working as LaTeX in Jupyter notebooks. Falls back to clean plain text in scripts or terminals.

Built from analysis of 3 years of PolyU AMA3724 past exam papers.

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Install

pip install furthermathskit

Quick Start

from furthermathskit import *

toolkit_menu()   # see all functions
quick_ref()      # formula sheet

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Functions

Linear Algebra

# Column-sum norm and spectral norm
matrix_norms([[-1,7,3,-3],[7,-1,3,-3],[3,3,-1,-7],[-3,-3,-7,-1]])

# Orthogonal projection of b onto col(A) + least squares
orthogonal_projection(b=[8,0,2,-2], A_list=[[...]])

# Change of basis matrix P_{B<-E} and coordinate vector [X]_B
change_of_basis(E, B, X)

# Jordan form with eigenspace dimension analysis + generalised eigenvector hints
jordan_analysis([[2,1,0,0,0],[0,2,0,0,0],[0,0,3,1,0],[0,0,0,3,1],[0,0,0,0,3]])

# Full SVD decomposition
svd_analysis([[1,2],[2,4]])

ODEs

Functions marked # are safe for exam questions that explicitly ban dsolve — they work purely via eigenvalues, integrals, and manual substitutions.

# 2nd-order constant-coefficient IVP
# Automatically uses undetermined coefficients with printed working
solve_2nd_order_ivp(a=6, b=-1, c=-1, y0=10, dy0=0)
solve_2nd_order_ivp(a=1, b=-4, c=3, y0=2, dy0=1, forcing=exp(5*t))

# Cauchy-Euler  a2 t^2 y'' + a1 t y'' + a0 y = 0
# Handles distinct, repeated, and complex roots                          #
solve_cauchy_euler(1, -3, 4)
solve_cauchy_euler(1, -3, 4, y0=1, dy0=2)

# System X'=AX via eigenvalue method (no dsolve)                        #
solve_system_eigenvalue([[0,1,0],[0,0,1],[-15,-23,-9]], t0=0, X0_list=[1,1,1])

# Euler-type system tX'=AX  (solutions are t^lambda * v)                #
solve_system_eigenvalue(A, t0=1, X0_list=[1,1,1], euler_type=True)

# Variation of parameters (no dsolve)                                    #
variation_of_parameters(y1=x*exp(x), y2=x, forcing_expr=2*x**2)

# Exact ODE check + solve for potential function F(x,y)=C               #
check_exact_ode(M_expr=cos(y), N_expr=y**3 - x*sin(y))

# Bernoulli ODE  y' + P(x)y = Q(x)y^n                                   #
solve_bernoulli(n_val=4, P_expr=-1, Q_expr=-exp(-x))

Market Models

# Solow growth model  k' + lambda*k = s*k^alpha  (Bernoulli)
solow_growth_model(alpha=Rational(3,11), s_val=Rational(1,3),
                   lam_val=Rational(1,4), k0=1)

# Market clearing equilibrium ODE
P = Function('P')
Qd = 5*P(t) - 4*P(t).diff(t) + P(t).diff(t,2)
Qs = 2*P(t) + exp(5*t) + 2
market_clearing_equilibrium(Qd, Qs, P0=2, dP0=1)

Fourier / PDE

# Heat IBVP — Neumann BCs (cosine series)
# Smart: if IC is a sum of cosines, uses direct mode matching
# No integration needed — just reads off coefficients
solve_heat_ibvp_neumann(k_val=7, L_val=9, ic_expr=cos(6*pi*x))

# With reaction term  ut = k uxx + a u  ->  substitution u = e^(at) w
solve_heat_ibvp_neumann(k_val=7, L_val=9, ic_expr=cos(6*pi*x),
                        extra_term="5u", change_var=(5, "u=e^(5t)w -> wt=7wxx"))

# Heat IBVP — Dirichlet BCs (sine series)
solve_heat_ibvp_dirichlet(k_val=5, L_val=pi, ic_expr=sin(3*x))

# Wave IBVP
solve_wave_ibvp(L_val=pi, ic_u_expr=sin(4*x)+2*sin(6*x),
                ic_ut_expr=2*sin(10*x))

# Fourier transform PDE on (-inf, inf) — guided steps
fourier_transform_pde({'a': 5, 'b': 3, 'c': 8, 'ic': 'sin(3 pi x)'})

Gram-Schmidt in L²([a,b])

# Orthonormalise a polynomial basis over [0,1]
basis = [S(1), x, x**2, x**3, x**4]
ONB   = l2_gram_schmidt(basis, a=0, b=1)

# Project a function onto the ONB
coeffs, reconstruction = l2_project(-10*x**3 + x**4, ONB)

# Raw inner product
l2_inner_product(x, x**2, a=0, b=1)   # -> 1/4

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Output style

In Jupyter notebooks — all expressions render as proper LaTeX:

• Section headers: styled HTML blocks
• Step-by-step working shown at each stage
• Final answers in a green bordered box
• Eigenvalues, matrices, solutions all rendered via display(Math(...))

In scripts / terminals — clean Unicode fallback with box-drawing characters.

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Topics covered

Topic	Notes
Matrix norms (column-sum, spectral)	Via eigenvalues of AᵀA
Orthogonal projection & least squares	Normal equations + null space
Change of basis	P_{B←E} and [X]_B
Jordan form	With generalised eigenvector hints
SVD	Full U, Σ, Vᵀ decomposition
2nd-order ODE IVP	Undetermined coefficients with working
Cauchy-Euler ODE	All root cases including complex
System of DEs (X'=AX)	Eigenvalue method, no dsolve
Euler-type system (tX'=AX)	t^λ solutions
Variation of parameters	Full Wronskian working, no dsolve
Exact ODE	Check + solve for F(x,y)
Bernoulli ODE	w=y^(1-n) substitution
Market equilibrium ODE	Arbitrary order
Solow growth model	Bernoulli reduction
Heat IBVP Neumann	Smart cosine mode matching
Heat IBVP Dirichlet	Sine series
Heat with reaction term	e^(at)·w substitution trick
Wave IBVP	aₙ and bₙ coefficients
Fourier transform PDE	Guided procedure + formula
Gram-Schmidt L²([a,b])	Full orthonormalisation
L² projection	Coefficient computation + reconstruction

License

MIT