Unit 6 of 10
Ordinary and partial differential equations
Exactly 7 in both papers. The same ideas repeat: singular solutions and PDE types.
- 7questions expected next time
- 7asked in 2025
- 7asked in 2026
- 4ideas asked in both papers
Clairaut, IVP uniqueness, PDE type and CF/PI were all asked in both papers.
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.
Singular solution of a Clairaut-type equation
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For y = px + f(p), eliminate p from y = px + f(p) and x + f′(p) = 0.
IVP: one, none or infinitely many solutions
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Lipschitz in y gives exactly one. y′ = y^(1/3), y(0) = 0 has infinitely many.
Classify a second-order PDE
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For Au_xx + 2Bu_xy + Cu_yy: B² − AC > 0 hyperbolic, = 0 parabolic, < 0 elliptic.
CF and PI of a constant-coefficient linear PDE
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Each factor (D − mD′) gives φ(y + mx) in the CF. For e^(ax + by), put D = a and D′ = b.
Asked in 2026
7 questions. Tap one for the answer and working.
Q41(y-px)^2=p^2(1+p^2): nature of singular solution relative to general solutionB
Answer: envelope of general solution
Equation is Clairaut: y = px +/- p*sqrt(1+p^2). Singular solution exists (from x = -f'(p)) and is the envelope of the family of lines.
Sure
Q51U_tt - c^2 u_xx transforms to canonical form using which methodB
Answer: characteristic variables
Hyperbolic u_tt - c^2 u_xx=0: xi=x-ct, eta=x+ct gives u_xi_eta=0.
Sure
Q57Eigenvalues of y''+lambda y=0, y(0)=y(pi)=0C
Answer: n^2
y=sin(nx), lambda=n^2, n=1,2,...
Sure
Q66Y'=y^(1/3), y(0)=0: unique, none, infinitely many, or exactly two solutionsC
y'=y^(1/3), y(0)=0: y=0 and y=±(2x/3)^(3/2) all work, and can be glued at any x0 (non-Lipschitz at 0) -> infinitely many solutions
Sure
Q74Classify PDE x^2 u_xx + 2xy u_xy + ... (elliptic/hyperbolic/parabolic/mixed)C
A=x^2, B=xy, C=y^2: B^2-AC = x^2y^2 - x^2y^2 = 0 -> parabolic
Sure
Q79(D^2-4D+4)z=0: form of complementary functionC
(D-2)^2 z=0 -> z=(A+Bx)e^(2x) (A,B arbitrary functions of y)
Sure
Q91Y'' + (1/x)y' type equation for x>0: evaluate W(x)*x (constant, x, x^2, 1/x) (OCR-uncertain)A
Homogeneous y'+y/x=0 -> y1=c/x, Wronskian (Abel) W=c/x so W*x=constant
Sure
Asked in 2025
7 questions. Tap one for the answer and working.
Q54PI for RHS e^{2x} sin xC
PI = e^{2x} (1/(D^2+2D+5)) sin x = e^{2x}(4 sin x - 2 cos x)/20 = -(1/10)e^{2x}(cos x - 2 sin x).
Matches the official key
Q55PI of a linear PDEB
z = x^2 y: z_xx = 2y, z_y = x^2, difference 2y - x^2.
Matches the official key
Q56IVP has unique/no/infinite solutionsB
General solution y = C x^4 gives y(0)=0 != 2: no solution.
Matches the official key
Q57Singular solution of Clairaut-type equationD
p-discriminant of p^3 - 4xy p + 8y^2: 256x^3y^3 - 1728y^4 = 0 => 27y = 4x^3; verified it satisfies the ODE.
Matches the official key
Q58General solution form of a given ODEB
Laplace equation: u = f(x+iy)+g(x-iy).
Matches the official key
Q59Match PDEs with elliptic/parabolic/hyperbolicD
a heat = parabolic (II); b wave = hyperbolic (III); c Laplace = elliptic (I); d y u_x - x u_y = 0 => u=f(x^2+y^2) (IV).
Matches the official key
Q60Complete solution of z = f(p,q)B
Clairaut form z = px+qy+f(p,q): complete integral z = ax+by+a^2+b^2.
Matches the official key
In the syllabus, not asked yet
Give each of these one revision pass. Skipped topics often turn up later.
- Separation of variables for Laplace, heat and wave
- Lagrange linear PDE (Pp + Qq = R)
Full syllabus for this unit
Existence and uniqueness for first-order IVPs, singular solutions, linear ODE theory, Sturm–Liouville; Lagrange and Charpit methods, constant-coefficient PDEs, classification of second-order PDEs, separation of variables.