Math Problems Vault

Problem 6128

The functions $y(x)$ and $z(x)$ satisfy the equation

y^{2}+y'^{2} = z'.

Moreover, it is given that

y(0) = a+b, \quad y(n) = ae^{n}+be^{-n}, \quad z(0) = a^{2}-b^{2}.

Find $y$ and $z$ from the condition that $z(n)$ is a maximum.

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Problem 6129

Consider the problem of the shortest distance between two points $M_{1}$ and $M_{2}$ in space as a Mayer problem.

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Problem 6130

Among all curves joining the given points $M_{1}(x_{1},y_{1})$ and $M_{2}(x_{2},y_{2})$ , lying in the vertical plane $XOY$ ( $OY$ --- downward), find the one along which, moving, a heavy point will arrive from $M_{1}$ to $M_{2}$ in the shortest time. The initial speed is $v_{1}$ ; besides gravity, the point is acted on by a resistance $R=kv^{n}$ .

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Problem 6131

Among the curves joining the given points $M_{1}(x_{1},y_{1})$ and $M_{2}(x_{2},y_{2})$ , lying in the plane $XOY$ ( $OY$ --- downward), find the one along which, moving, a point will arrive at $M_{2}$ with the greatest speed. Here the speed of the point at $M_{1}$ equals $v_{1}$ ; the point is acted on by gravity and by a resistance force $R=kv^{n}$ . Compare the normal reaction of the curve with the centrifugal force.

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Problem 6132

Join the points $M_{1}(0,0,0)$ and $M_{2}(a,0,0)$ by a curve having given curvature $\frac{1}{\rho }$ and the greatest length.

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Problem 6133

An airplane departs from the point $A(x_{1},y_{1})$ at the moment $t=0$ , intending to reach the point $B(x_{2},y_{2})$ in the shortest time. Find how the airplane must be directed for this, given that the distribution of wind in space and time is known, i.e., the components of the wind velocity $w$ along the axes $OX$ and $OY$ are known: $w_{x}=u(x,y,t)$ , $w_{y}=v(x,y,t)$ . The airplane's own speed is $V$ , and $w^{2}=u^{2}+v^{2} < V^{2}$ .

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Problem 6134

An airplane wing of length $l$ is located along the axis $OX$ and moves parallel to the axis $OY$ with speed $v_{0}$ (the plane $XOY$ is horizontal). Denote by $f(x) \, dx$ the lift of an element of the wing of length $dx$ . In aerodynamics it is shown, under certain assumptions, that under these conditions a vortex of intensity $d\Gamma = \frac{1}{\rho v_{0}}f'(x) \, dx$ is shed from the element of the wing. Here $\rho$ is the density of air, taken to be constant. (This formula is a consequence of the basic formula of N.E.Zhukovsky.) These vortices induce at every point $\xi$ of the wing a downward air velocity

w(\xi ) = \frac{1}{4\pi \rho v_{0}} \int _{-l/2}^{l/2} \frac{f'(x)}{\xi -x} \, dx.

Here the axis $OZ$ is directed downward, and the integral is understood everywhere, here and below, in the principal-value sense. The induced drag of the wing is determined by the formula

B = \int _{-l/2}^{l/2} \frac{\omega (\xi )}{v_{0}}f(\xi ) \, d\xi = \frac{1}{4\pi \rho v_{0}^{2}} \int _{-l/2}^{l/2}\int _{-l/2}^{l/2} \frac{f'(x)f(\xi )}{\xi -x} \, dx \, d\xi .

It is required to determine, for a given total lift of the wing

A = \int _{-l/2}^{l/2} f(x) \, dx

the distribution of lift, i.e., the function $f(x)$ , for which the induced drag is smallest. Here it should be taken into account that, because of the absence of vortices outside the interval $-\frac{l}{2} \leq x \leq \frac{l}{2}$ , the condition $f\left(\pm \frac{l}{2}\right) = 0$ must hold.

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Problem 6135

Two equations are given

u' = f(x,y,y',z,u), \quad z' = g(x,y,y',z,u).

Find $y$ as a function of $x$ under the condition that the values of $x$ , $y$ , $z$ , and $u$ are given at $x=x_{0}$ , and $y$ is also given at $x=x_{1}$ , while $u(x_{1})$ must be an extremum. Example:

u(x_{1}) = \int _{x_{0}}^{x_{1}} f(x,y,y',z) \, dx; \quad z = \int _{x_{0}}^{x} \sqrt{1+y'^{2}} \, dx.

Form Euler's differential equations.

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Problem 6136

Find the extremum of the function $W = f(u,v)$ , where

u = \int _{t_{1}}^{t_{2}} F(x,y,x',y') \, dt, \quad v = \int _{t_{1}}^{t_{2}} G(x,y,x',y') \, dt

for given values of $x$ and $y$ at the ends of the interval $(t_{1},t_{2})$ .

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Problem 6137

Find the form of the extremals for the problem of the minimum of the integral

\int _{A}^{B} y \, dx \cdot \int _{A}^{B} \sqrt{1+y'^{2}} \, dx,

where $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$ are given points.

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