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CSIR NET Part C

Counterexample bank

Part C is won by knowing which tempting claims are false. 149 counterexamples; 86 free to read right now. Pick your subject below.

Knowing the claim is false is half of it. The mistakes that cost marks shows the wrong options candidates actually pick, and the reasoning slip behind each one.

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Tagged “ODE”

“Every IVP has a unique solution”

False — y′ = y1/3y^{1/3}, y(0) = 0

“A continuous right-hand side gives a unique solution”

False — y′ = y1/3y^{1/3}, y(0) = 0

“A locally unique solution exists for all time”
“W(f, g) ≡ 0 implies f and g are linearly dependent”

False — f(x)=x2,g(x)=xf(x) = x^{2}, g(x) = x|x| on R\mathbb{R}

“Every boundary value problem has a Green's function”

False — y′′+π2y=fy'' + \pi^{2}y = f on [0,1] with y(0) = y(1) = 0

“Linearisation determines stability at every equilibrium”

False — ẋ =−y−x3= -y - x^{3}, ẏ =x−y3= x - y^{3} at the origin