Classify the following statements as true or false. Give a short proof of a counter example in support of your answer.

i) The solution of the differential equation    exists, but is not unique.

ii) The differential equation representing all tangents ty = x + t²  at the point (t², 2t) to the parabola y² = 4x is   x(y')² + yy'+1=0 

iii) The p.d.e. au_xx + 2b u_xy + cu,_yy = 0 where a, are constants is irreducible when b² ac = 0 

iv) The functions f₁(x) = cos²x, f₂(x) = sin²x, f₃(x) = sec² x and  f₄(x) = tan²x  are linearly dependent on the interval ] - π/2, π/2 [ 

v) The solution of the second order partial differential equation   involves two arbitrary constants.

i) True
By the existence and uniqueness theorem, a solution exists but may not be unique if the function or its partial derivative with respect to yy is not continuous. For example, the equation y′=∣y∣1/3y' = |y|^{1/3} has infinitely many solutions passing through (0,0)(0,0).

ii) True
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