Unit 7 of 10
Calculus of variations, integral equations, transforms
Grew from 5 to 8. Integral equations alone went from 2 questions to 6.
- 5–8questions expected next time
- 5asked in 2025
- 8asked in 2026
- 2ideas asked in both papers
Separable kernels and Volterra-by-Laplace are short and now asked often.
Most likely questions
Practise each one until you can do it without notes, then tick it. Ticks stay in this browser.
Already repeated
These came up in 2025 and again in 2026. Answer each in your head, then tap to check.
Euler–Lagrange equation and extremals
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∂F/∂y − d/dx (∂F/∂y′) = 0. If F has no x, then F − y′ ∂F/∂y′ is constant.
Solve a Fredholm or Volterra integral equation
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A separable kernel turns it into a small linear system. A Volterra convolution kernel is solved by Laplace transforms.
Asked in 2026
8 questions. Tap one for the answer and working.
Q32Symmetric degenerate kernel of rank 2: max number of nonzero eigenvalues, eigenfunction orthogonalityD
Answer: 2, orthogonal
Rank-2 symmetric degenerate kernel has at most 2 nonzero eigenvalues; eigenfunctions of a symmetric kernel for distinct eigenvalues are orthogonal.
Sure
Q39Y(x)=1+lambda*integral of x t y(t) dt on [0,1]: nonzero eigenvalue via consistency conditionD
Answer: lambda=3
y=1+lambda*c*x with c=int t y dt; c=1/2+lambda*c/3 so c(1-lambda/3)=1/2; singular (characteristic value) at lambda=3.
Sure
Q45BVP converted via Green's function to symmetric Fredholm equation; smallest lambda where resolvent kernel failsA
Answer: pi^2
EL: y''=lambda*y, Dirichlet Green's function of -d^2/dx^2 has characteristic values n^2 pi^2; smallest is pi^2 (sign convention aside, only positive options).
Fairly sure
sign of lambda (true value -pi^2 under one convention); chose smallest positive characteristic value pi^2.
Q68Evaluate improper integral of sin x / x (type) using Fourier-transform reasoningD
Int_0^inf sin x / x dx = pi/2 (Dirichlet integral, via Fourier transform of rectangular pulse)
Sure
Q76Extremal of integral (y'^2+y^2) with constraint integral y=1 via Lagrange multiplier: formC
Euler-Lagrange: y''-y=-lambda/2 gives y=lambda/2 + c1 e^x + c2 e^-x; with free ends (natural BC y'(0)=y'(1)=0) c1=c2=0 so y=constant=1 (int y=1). Constant is the only option that can occur
Fairly sure
No boundary conditions are printed. C assumes natural boundary conditions.
Q82Degenerate kernel gives singular linear system: condition for existence of solution (rank condition)B
Singular (degenerate-kernel) system is solvable iff consistency holds: rank A = rank [A|b] (Fredholm alternative). "Rank condition satisfied" is the only sensible option
Sure
Q89Volterra kernel not of convolution type: correct way to apply Laplace transformB
Laplace transform needs convolution kernel; a variable transformation (e.g. t=e^s) can convert some non-convolution kernels (Mellin type) to convolution form. Other options (ignore kernel, convert to PDE, direct transform) are not valid approaches
Not sure
poorly posed; B (variable transformation) is a best guess.
Q97Y(x)=lambda * integral of G(x,t)y(t)dt: condition for nontrivial solution via operator theory (OCR-uncertain)D
Homogeneous Fredholm eqn y=lambda K y has non-trivial solution iff lambda is an eigenvalue i.e. Fredholm determinant D(lambda)=det(I-lambda K)=0
Sure
Asked in 2025
5 questions. Tap one for the answer and working.
Q74Extremal of a functional with end conditionsC
F has no y: y'/(x sqrt(1+y'^2)) = c gives x^2+(y-k)^2=R^2; through (1,0),(2,1): k=2, R^2=5
Matches the official key
Q75Solution of an integral equationA
L{t^n F(t)} = (-1)^n d^n f/ds^n
Matches the official key
Q76Solution of Fredholm integral equationA
c = int t phi = 1/2 + lambda c/3, c = 3/(2(3-lambda)); phi = 1 + 3 lambda x/(2(3-lambda))
Matches the official key
Q77Laplace transform of t cos tB
-d/ds[s/(s^2+1)] = (s^2-1)/(s^2+1)^2
Matches the official key
Q78Euler–Lagrange equation is 2nd-order ODE (type)C
E-L: F_y'y' y'' + F_y'y y' + F_y'x - F_y = 0 is second order, in general non-linear (depends on F)
Matches the official key
Both solves agree on C. B (linear) is the only real competitor.
In the syllabus, not asked yet
Give each of these one revision pass. Skipped topics often turn up later.
- Brachistochrone and catenary
- Neumann series
Full syllabus for this unit
Functionals, Euler–Lagrange, isoperimetric problems; Fredholm and Volterra equations, successive approximation, separable kernels, Green’s function; Laplace transforms.