Question

Please provide step by step solution.

(a) Express the Falkner-Skan equation 1 2 = -m, 11" + 3(m +1)yy" – m (4') with initial conditions y(0) = 0, y'(0) = 0, y"(0) = G, where m and G are constants, as an equivalent system of three first order ordinary differential equations. (b) Are there any intervals for which the conditions for the Existence-Uniqueness Theorem for systems of equations are not satisfied for the Falkner-Skan equation (and its asso- ciated system) here? If so, identify the key elements of such intervals, along with the conditions that are violated.

Answer 1

Answer:-

a) Falkner. skan equation is

${y}^{'''}+\frac{1}{2}(m+1)y{y}''-m({y}')^{2}=-m$...................................(1)

with  $y(0)=0,$$y{}'(0)=0,$   $y{}''(0)=G,$

Where m,G are constants.

Let

$x_{1}=y$

$x_{2}=y{}'$

$x_{3}=y{}''$

So from (1) , we get

$x_{1}{}'=x_{2}$

$x_{2}{}'=x_{3}$

$x_{3}{}'=-m-\frac{1}{2}(m+1)x_{1}x_{3}+mx_{2}^{2}$

With  $(x_{1}(0),x_{2}(0),x_{3}(0))=(0,0,G)$

i.e . if we take   $X=(x_{1},x_{2},x_{3})$. Then

$X{}'=f(x)$

$X(0)=(0,0,G)$..........................................(2)

Where $f(x)=f(x_{1},x_{2},x_{3})$

  $=(x_{2},x_{3},-m-\frac{1}{2}(m+1)x_{1}x_{3}+mx_{2}^{2})$

$\therefore$ (2) is the required system of , 1st order $\triangle$ DE, equivalent to (1)

b) Clearly, as  $f(x_{1},x_{2},x_{3})=(x_{2},x_{3},-m-\frac{1}{2}(m+1)x_{1}x_{3}+mx_{2}^{2})$ has each component is smooth (being polnomial in $(x_{1},x_{2},x_{3})$ . so with any given initial condition , (2) ( or associated (1) ) has locally unique solution around O. (By existence uniqueness theorem ). and because of the same reason, there are no such intervals for which, the conditions of existence unidueness theorom are violated.

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