Question

Find a parametric solution of x^2+y^2 = 0 as follows.

(a) Write an equation for y in terms of x and show that
y = √ 2+ 1 .
(b) Using as the independent variable, arrange this as a linear first-order equation for .
(c) Find an appropriate integrating factor to obtain y=1n(x-1)+ (1-x)^2, which, together with the expression for obtained in (a), gives a parameterisation of the solution.
(d) Reverse the roles of and in steps (a) to (c), putting x= −1, and show that essentially the same parameterisation is obtained

Answer 1

The task to find a parametric solution for the equation x^2 + y^2 = 0 leads to the trivial solution where both x and y are zero. Given the equation's symmetry, reversing the roles of x and y yields the same parametric solution.

Step-by-step explanation:

The question involves finding a parametric solution to the equation x2 + y2 = 0. However, the equation provided already suggests that the only possible real solutions for x and y are when both are zero since no real number squared can yield a negative result. Therefore, the presumption of having a solution such as y = √(2 + 1) appears to be a typo or mistake, as this equation is not equivalent to x2 + y2 = 0.

Typically, the process of finding a parametric solution involves expressing both variables x and y in terms of a third variable, often denoted as t (parameter). However, for the equation x2 + y2 = 0, the parametric solution can be trivially given by x(t) = 0, y(t) = 0 for all values of t.

For part (b), one would typically rearrange the equation into a linear first-order equation. However, as stated, this cannot be done for the given equation unless considering complex numbers. For part (c), no integrating factor is relevant because the solution is already known. As for part (d), reversing the roles of x and y yields no difference due to the symmetry of the equation.