Math Problems Vault

Problem 5342

Find the power-series expansion of the general integral of the equation $y' - xy = 1$ .

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Problem 5343

Find the power-series expansion of the general integral of the equation $(1+x)y' - ny = 0$ .

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Problem 5344

Find the power-series expansion of the general integral of the equation $y'' + y = 0$ .

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Problem 5345

Find the power-series expansion of the general integral of the equation $y'' - xy = 0$ .

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Problem 5346

Prove that the corresponding particular integral of the differential equation

x(x-1)y'' + \left((\alpha +\beta +1)x-\gamma \right)y' + \alpha \beta y = 0

can be represented by the indicated power series:

y = 1 + \frac{\alpha \cdot \beta }{1\cdot \gamma }x + \frac{\alpha (\alpha +1)}{1\cdot 2}\cdot \frac{\beta (\beta +1)}{\gamma (\gamma +1)}x^{2} + \frac{\alpha (\alpha +1)(\alpha +2)}{1\cdot 2\cdot 3}\cdot \frac{\beta (\beta +1)(\beta +2)}{\gamma (\gamma +1)(\gamma +2)}x^{3} + \ldots

(Gauss's hypergeometric series).

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Problem 5347

Prove that the corresponding particular integral of the differential equation

x^{2}y'' + xy' + (x^{2}-n^{2})y = 0

can be represented by the indicated power series:

y = \sum _{\nu =0}^{\infty } \frac{(-1)^{\nu }\left(\frac{x}{2}\right)^{n+2\nu }}{\nu !\, \Gamma (n+\nu +1)}

(the Bessel function $J_{n}(x)$ ).

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Problem 5348

Prove that the corresponding particular integral of the differential equation

x(x^{4}-1)u'' - (3x^{4}-1)u' + x^{3}u = 0

can be represented by the indicated power series:

u = 1 + \left(\frac{1}{4}\right)^{2}x^{4} + \left(\frac{1\cdot 5}{4\cdot 8}\right)^{2}x^{8} + \left(\frac{1\cdot 5\cdot 9}{4\cdot 8\cdot 12}\right)^{2}x^{12} + \ldots ; \qquad \left|x\right| < 1.

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Problem 5349

Prove that the corresponding particular integral of the differential equation

x^{2}(x^{4}-1)v''' + 3x(3x^{4}-1)v'' + (19x^{4}-1)v' + 8x^{3}v = 0

can be represented by the indicated power series:

v = 1 + \left(\frac{1}{2}\right)^{3}x^{4} + \left(\frac{1\cdot 3}{2\cdot 4}\right)^{3}x^{8} + \left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)^{3}x^{12} + \ldots ; \qquad \left|x\right| < 1.

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Problem 5350

Prove Gauss's equality:

\left(1 + \left(\frac{1}{4}\right)^{2}x^{4} + \left(\frac{1\cdot 5}{4\cdot 8}\right)^{2}x^{8} + \left(\frac{1\cdot 5\cdot 9}{4\cdot 8\cdot 12}\right)^{2}x^{12} + \ldots \right)^{2} = 1 + \left(\frac{1}{2}\right)^{3}x^{4} + \left(\frac{1\cdot 3}{2\cdot 4}\right)^{3}x^{8} + \left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)^{3}x^{12} + \ldots ; \qquad \left|x\right|<1.

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Problem 5351

Prove that, applying Picard's method to the equation $y' = y$ , with $a=0$ , $b=1$ , we obtain:

y = 1 + x + \frac{x^{2}}{2!} + \ldots + \frac{x^{n-1}}{(n-1)!} + \frac{x^{n}}{n!} + \ldots

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Remark.

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Picard's method of finding the integral of the equation $y' = \varphi (x,y)$ , which equals $b$ at $x=a$ , consists in successively constructing a sequence of functions $y_{0}, y_{1}, y_{2}, \ldots$ by the formula $y_{n} = b + \displaystyle \int_{a}^{x} \varphi (x,y_{n-1})\, dx$ ; $y_{0}=b$ . If $y_{n} \to y$ as $n \to \infty$ , then $y$ is the desired integral.

Problem 5352

Prove that in the preceding example (problem 5351), $y_{n} \to y$ as $n \to \infty$ for an arbitrary choice of the function $y_{0}$ .

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Problem 5353

Prove that, replacing the equation

y'' + p(x)y' + q(x)y = 0

by the system of equations

y' = u; \qquad u' = -p(x)u - q(x)y

and applying Picard's method to solve these equations, we obtain a convergent process if the radius vector from the origin to the point $x$ does not pass, in the complex plane of the variable $x$ , through any singular point of the functions $p(x)$ and $q(x)$ .

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Problem 5354

Solve the equation $y' - y = axy$ by the method of successive approximations, setting $y_{0}=0$ and computing the quantities $y_{n}$ by the formula $y_{n}' - y_{n} = axy_{n-1}$ (at $x=0$ we must have $y=C$ ).

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Problem 5355

In the same way (see problem 5354), solve the equation $y'' + a^{2}y = \sigma xy$ , where $\sigma$ is a small parameter. Here $y=0$ , $y'=a$ at $x=0$ . It is useful to apply the theorem: if $y''+a^{2}y=f(x)$ , then

y = C_{1}\sin ax + C_{2}\cos ax + \frac{1}{a}\int _{0}^{x} f(t)\sin a(x-t)\, dt.

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Problem 5356

Using the substitution $x = n + \dfrac {1}{\sqrt{n}}$ , show that the integral of the equation $y'' + xy = 0$ on the interval $(n, \pi +n)$ , where $n$ is large, has approximately $\sqrt{n}$ roots.

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Problem 5357

Prove that the integral of Bessel's equation

x^{2}y'' + xy' + (x^{2}-n^{2})y = 0

on an interval $(a, a+b)$ far from the origin has approximately $\dfrac {b}{\pi }$ roots.

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Problem 5358

The integral of the equation $y'' + y = \sigma y'^{2}$ has initial conditions: $x=0$ , $y=1$ , $y'=0$ . Find the first three terms of its expansion in powers of the small parameter $\sigma$ .

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Problem 5359

The Chebyshev--Hermite polynomials $H_{n}(x) = \dfrac {e^{\frac{x^{2}}{2}}}{n!}\dfrac {d^{n}e^{-\frac{x^{2}}{2}}}{dx^{n}}$ satisfy the equation

H_{n}''(x) - xH_{n}'(x) + nH_{n}(x) = 0.

By the substitution $H_{n}(x) = f(x)\cdot y$ , transform this equation to the form

y'' + \left(-\frac{x^{2}}{4} + n + \frac{1}{2}\right)y = 0.

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Problem 5360

Find the asymptotic expression, as $n \to \infty$ , for the Chebyshev--Hermite polynomials $H_{n}(x) = \dfrac {e^{\frac{x^{2}}{2}}}{n!}\dfrac {d^{n}e^{-\frac{x^{2}}{2}}}{dx^{n}}$ (see problem 5359).

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Problem 5361

Find, for $0<x<\pi$ , the integral of the equation

y'' + 4y = \pi x - x^{2},

by expanding it in a series of the form $\sum a_{n}\sin nx$ . Boundary conditions: $y=0$ at $x=0$ and $x=\pi$ .

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Problem 5362

Find the integral of the equation $y^{\mathrm{IV}} + y = \pi - \left|x\right|$ for $-\pi <x<\pi$ and boundary conditions $y' = y''' = 0$ at $x = \pm \pi$ .

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Problem 5363

Construct a table of values of $y$ at intervals of $0{,}05$ , given $y'+2xy=0$ and $y=1$ at $x=0$ . Here the values $y_{-2}, y_{-1}, y_{0}, y_{1}, y_{2}$ are to be computed by means of a series expansion, in the Adams formula terms up through the third differences are to be retained, and the computations are to be carried out to the sixth decimal digit.

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Remark.

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Numerical integration of the differential equation $y'=f(x,y)$ consists in this: given the initial condition $y=y_{0}$ at $x=x_{0}$ , one constructs a table of values of $y$ for values of $x$ equal to the numbers $x_{0}, x_{0}+h, x_{0}+2h, \ldots$ , where $h$ is a chosen number, usually fairly small.

The Adams--Störmer formula: $\Delta y_{n} = u_{n} + \dfrac {1}{2}\Delta u_{n-1} + \dfrac {5}{12}\Delta^{2}u_{n-2} + \dfrac {3}{8}\Delta^{3}u_{n-3} + \dfrac {251}{720}\Delta^{4}u_{n-4} + \dfrac {1}{3}\Delta^{5}u_{n-5} + \ldots$ , where $u = hf(x,y)$ ; this formula makes it possible to compute $y_{n+1}$ .

Problem 5364

Find by the Adams--Störmer method, at $x=2$ , the integral of the equation $y' = x+y$ that equals unity at $x=1$ ; here use the formula for $\Delta y$ taken up through the fourth differences inclusive, and carry out the computations to the fifth digit.

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Problem 5365

Find a table of values of the integral of the equation

y' = \sqrt{x} + \sqrt{y} \quad \text{for} \quad x = 0;\ 0{,}1;\ 0{,}2;\ \ldots \ 1,

given $y=0$ at $x=0$ .

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Remark.

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The expansion of $y$ near the start has the form $y = \dfrac {2}{3}x^{\frac32} + \dfrac {4}{7}\sqrt{\dfrac {2}{3}}x^{\frac74} + \dfrac {1}{7}x^{2} + \dfrac {1}{49}\sqrt{\dfrac {2}{3}}x^{\frac94} + \ldots$ It is advantageous to introduce a new variable by setting $x = \xi^{4}$ .

Problem 5366

Construct a table of values of the functions $\sin x$ and $\cos x$ at intervals of $0{,}1$ from $0$ to $1$ , using the equations $(\sin x)' = \cos x$ ; $(\cos x)' = -\sin x$ and applying the Adams--Störmer method.

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Problem 5367

Construct a table of values of the Bessel functions $J_{0}(x)$ and $J_{0}'(x)$ for values of $x$ at intervals of one tenth from zero to one, replacing the equation for $J_{0}(x)$ , which has the form

xJ_{0}''(x) + J_{0}'(x) + xJ_{0}(x) = 0,

by the system: $xv' + v + xu = 0$ ; $u' = v$ .

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Remark.

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For second-order equations of the form $y'' = f(x,y,y')$ one may use the formulas:

\begin{align} \Delta ^{2}y_{n-1} & = h^{2}f_{n} + \frac{1}{12}h^{2}\Delta ^{2}f_{n-2} + \frac{1}{12}h^{2}\Delta ^{3}f_{n-3} + \frac{19}{240}h^{2}\Delta ^{4}f_{n-4} + \frac{3}{40}h^{2}\Delta ^{5}f_{n-5}, \\ \Delta y_{n} & = hy_{n}' + \frac{h^{2}}{2}f_{n} + \frac{h^{2}}{6}\Delta f_{n-1} + \frac{h^{2}}{8}\Delta ^{2}f_{n-2} + \frac{19h^{2}}{180}\Delta ^{3}f_{n-3} + \frac{3h^{2}}{32}\Delta ^{4}f_{n-4}, \\ \Delta y_{n}' & = hf_{n} + \frac{h}{2}\Delta f_{n-1} + \frac{5}{12}h\Delta ^{2}f_{n-2} + \frac{3}{8}h\Delta ^{3}f_{n-3} + \frac{251}{720}h\Delta ^{4}f_{n-4}. \end{align}

Problem 5368

Construct a table of values, at intervals of $0{,}1$ on the interval $(0,1)$ , of the function $y = \operatorname {ch}x$ , satisfying the equation $y'' = y$ and the initial conditions: at $x=1$ , $y=1$ , $y'=0$ .

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Problem 5369

Construct a table of values of the function $J_{0}(x)$ for values of $x$ at intervals of $0{,}1$ , knowing that $J_{0}(x)$ satisfies the differential equation $xJ_{0}''(x) + J_{0}'(x) + xJ_{0}(x) = 0$ , with initial values at $x=0$ : $J_{0}(0)=1$ , $J_{0}'(0)=0$ .

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Problem 5370

Knowing the values: $\cos 1 = 0{,}54030$ ; $\cos 1{,}1 = 0{,}45360$ ; $\cos 1{,}2 = 0{,}36236$ ; $\cos 1{,}3 = 0{,}26750$ ; $\cos 1{,}4 = 0{,}16997$ , using the equation $y'' = -y$ , find the values of $\cos x$ at $x=1{,}5$ ; $1{,}6$ ; $1{,}7$ ; $1{,}8$ ; $1{,}9$ ; $2{,}0$ .

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