Math Problems Vault

Problem 5674

Solve, by Cauchy's method, the equation $y''-2y'+y = f(x)$ ; $y_{0} = a$ , $y_{0}' = b$ , given these initial conditions.

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Remark.

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Cauchy's method for finding the integrals of a linear system of differential equations with constant coefficients

\frac{dx_{m}}{dt}+\sum _{\nu =1}^{n} \alpha _{m\nu }x_{\nu } = \varphi _{m}(t), \quad m = 1,2,\ldots ,n \tag {*}

consists in the following:

Introducing the symbol $\delta_{\mu \nu } = 0$ for $\mu \neq \nu$ and $\delta_{\nu \nu } = 1$ , we solve the equations
$\sum _{\nu =1}^{n} (\alpha _{\mu \nu }+s\delta _{\mu \nu })A_{\nu } = \xi _{\mu }; \quad \mu = 1,2,\ldots ,n; \tag {**}$
which gives:
$A_{\nu } = \frac{\sum _{\mu =1}^{n} B_{\nu \mu }(s)\xi _{\mu }}{\Delta (s)},$
where $\Delta (s)$ is the determinant of the system $(**)$ .
We form the functions
$\omega _{m}(s) = \sum _{\mu =1}^{n} B_{m\mu }(s)\left(\xi _{\mu }+\int _{0}^{t} \varphi _{\mu }(\tau )e^{-s\tau }\, d\tau \right)e^{s\tau }.$

Then the equations $(*)$ have the system of particular integrals:

x_{m} = \frac{1}{2\pi i}\int _{C} \frac{\omega _{m}(s)}{\Delta (s)}\, ds, \quad m = 1,2,\ldots ,n.

Here the integration is taken in the positive direction around the circle $\left|s\right| = R$ , large enough that all the roots of the equation $\Delta (s) = 0$ lie inside the circle. At $t = 0$ the quantities $x_{m}$ reduce to $\xi_{m}$ .

For a single equation

y^{(n)}+a_{1}y^{(n-1)}+\ldots +a_{n}y = f(x)

Cauchy's rule becomes the following:

We form the function
$\varphi (s) = \sum _{\nu =0}^{n-1} y_{0}^{(\nu )}\psi _{\nu }(s),$
where $\psi_{\nu }(s) = s^{n-\nu -1}+a_{1}s^{n-\nu -2}+\ldots +a_{n-\nu -1}$ .
Then
$y = \frac{1}{2\pi i}\int \frac{\sum _{\nu =0}^{n-1} y_{0}^{(\nu )}\psi _{\nu }(s)\cdot e^{sx}+\int _{0}^{x} e^{(x-t)s}f(t)\, dt}{s^{n}+a_{1}s^{n-1}+\ldots +a_{n}}\, ds.$

Here $y_{0}, y_{0}', \ldots , y_{0}^{(n-1)}$ are the values at $x = 0$ , and the integration proceeds around the circle $\left|s\right| = R$ , inside which lie all the roots of the denominator.

Problem 5675

Solve, by Cauchy's method (problem 5674), the system of equations $x'-x+y = 0$ , $y'+x+z = 0$ , $z'-x+y = 0$ ; $x_{0} = a$ , $y_{0} = b$ , $z_{0} = c$ , given these initial conditions.

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Problem 5676

Solve, by Cauchy's method (problem 5674), the system of equations $x'+x+y = e^{t}$ , $y'-x+3y = e^{-t}$ ; $x_{0} = -y_{0} = 1$ , given these initial conditions.

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Problem 5677

Solve, by Cauchy's method (problem 5674), the system of equations $x' = 4x+2y+2z$ , $y'+x+z = 0$ , $z'+2x+y = 0$ ; $-x_{0} = y_{0} = z_{0} = 1$ , given these initial conditions.

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Problem 5678

Solve, by Cauchy's method (problem 5674), the system of equations $x'' = 3x+4y$ , $y''+x+y = 0$ ; $x_{0} = y_{0}' = 1$ , $y_{0} = x_{0}' = 0$ , given these initial conditions.

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Problem 5679

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

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Problem 5680

Find the general integral by means of power series for the equation $y''-x^{2}y = 0$ .

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Problem 5681

Find the general integral by means of power series for the equation $y'''+xy = 0$ .

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Problem 5682

Find the general integral by means of power series for the equation $y'''+x^{3}y = 0$ .

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Problem 5683

Find the general integral by means of power series for the equation $(1-x^{2})y''-4xy'-2y = 0$ .

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Problem 5684

Find the general integral by means of power series for the equation $xy''+y'+y = 0$ .

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Problem 5685

Find the general integral by means of power series for the equation $xy''+2y'+xy = 0$ .

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Problem 5686

Prove that the equation $(1-x^{2})y''-5xy'-4y = 0$ is satisfied by the series

\sum _{n=0}^{\infty } \frac{(n!)^{2}}{(2n)!}(2x)^{2n} \quad \text{and} \quad \sum _{n=0}^{\infty } \frac{(2n+1)!}{(n!)^{2}}\cdot \left(\frac{x}{2}\right)^{2n+1},

the first of which equals $\dfrac {1}{2}\left((\arcsin x)^{2}\right)''$ .

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Problem 5687

Prove that the general integral of the equation

(1-x^{2})y''-(\alpha +\beta +1)xy'-\alpha \beta y = 0

equals $C_{1}y_{1}+C_{2}y_{2}$ , where

y_{1} = 1+\sum _{n=1}^{\infty } \frac{\alpha (\alpha +2)\cdots (\alpha +2n-2)\beta (\beta +2)\cdots (\beta +2n-2)}{(2n)!}x^{2n},

y_{2} = \sum _{n=0}^{\infty } \left(\frac{1}{(2n-1)!}(\alpha +1)(\alpha +3)\cdots (\alpha +2n-1)(\beta +1)(\beta +3)\cdots (\beta +2n-1)x^{2n+1}\right).

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Problem 5688

Prove that the equation $x^{2}y'+(x-1)y+1 = 0$ , whose general integral is

y = \frac{C}{x}e^{-\frac{1}{x}}-\frac{1}{x}e^{-\frac{1}{x}}\int xe^{\frac{1}{x}}\, dx,

is formally satisfied by the series $\sum_{n=0}^{\infty } n!x^{n}$ , which diverges for every $x \neq 0$ .

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Problem 5689

Find the integral of the equation

(1-x^{2})y''-2xy'+n(n+1)y = 0,

where $2n$ is a non-integer, as a series in positive powers of $x$ .

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Problem 5690

Find the integral of the equation $(1-x^{2})y''-2xy'+n(n+1)y = 0$ , where $2n$ is a non-integer (problem 5689), as a series in negative powers of $x$ .

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Problem 5691

Find the general integral of the equation

4x(x-1)y''+4\left((a+2)x-a-1\right)y'+(2a+1)y = 0,

representing it by the identity $y = C_{1}\varphi (x,a)+C_{2}x^{-a}\varphi (-a,x)$ , where $\varphi (a,x)$ is a series in powers of $x$ , equal to $1$ at $x = 0$ .

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Problem 5692

Find the integrals of the previous equation (problem 5691) for $a = 0$ as series converging in a neighborhood of the point $x = 1$ .

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Problem 5693

Find the integral of Bessel's equation

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

setting $y = C_{1}y_{1}+C_{2}y_{2}$ , where $y_{1}$ and $y_{2}$ are generalized power series of the form $x^{\sigma }(a_{0}+a_{1}x+a_{2}x^{2}+\cdots )$ , where $\sigma$ is some constant, not necessarily an integer. The number $n$ is assumed to be a non-integer.

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Problem 5694

Show that the general integral of Gauss's equation

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

can, for $\left|x\right| < 1$ , be represented by the formula

y = C_{1}F(\alpha ,\beta ,\gamma ;x)+C_{2}x^{1-\gamma }F(\alpha -\gamma +1,\beta -\gamma +1,2-\gamma ;x),

where $F(\alpha ,\beta ,\gamma ;x)$ denotes the hypergeometric series:

F(\alpha ,\beta ,\gamma ;x) = 1+\frac{\alpha }{1}\cdot \frac{\beta }{\gamma }x+\frac{\alpha (\alpha +1)}{1\cdot 2}\cdot \frac{\beta (\beta +1)}{\gamma (\gamma +1)}x^{2}+\ldots .

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Problem 5695

By means of the Laplace transform, integrate the equation $xy''+(2n+1)y'+xy = 0$ .

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Problem 5696

By means of the Laplace transform (problem 5695), integrate the equation $xy''-(\alpha +\beta )(1+x)y'+\alpha \beta xy = 0$ .

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Problem 5697

By means of the Laplace transform (problem 5695), integrate the equation $xy''+2ay'-q^{2}xy = 0$ ; $a > 0$ .

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Problem 5698

Prove that for $0 < a < 2$ the general integral of the equation $xy''+ay'-q^{2}xy = 0$ can be represented by the formula

y = \int _{0}^{\pi } e^{xq\cos \varphi }(A\sin ^{a-1}\varphi +Bx^{1-a}\sin ^{1-a}\varphi )\, d\varphi ,

and for $a = 1$ by the formula:

y = \int _{0}^{\pi } e^{xq\cos \varphi }\left(A+B\ln (x\sin ^{2}\varphi )\right)\, d\varphi .

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Problem 5699

Find two particular integrals of the equation $xy'''-y = 0$ , setting:

y_{1} = \int \sin \frac{x}{t}\cdot u(t)\, dt; \quad y_{2} = \int \cos \frac{x}{t}\cdot v(t)\, dt.

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Problem 5700

Find two pairs of particular integrals of the system

2xy'+2z'+y = 0; \quad 2y' = 2xz'+z; \quad \mathrm{Im}\left(x\right) < 1,

seeking one of the unknowns in the form of the integral $\int u(t)\sin xt\, dt$ , and the other in the form of the integral $\int v(t)\cos xt\, dt$ .

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Problem 5701

Prove that the integral of Legendre's equation

(1-x^{2})y''-2xy'+n(n+1)y = 0

can be represented in one of the forms:

y = \int _{0}^{1} t^{-\frac{n+1}{2}}(1-t)^{-\frac{n+2}{2}}(x^{2}-t)^{\frac{n}{2}}\, dt; \quad n > 0,

y = \int _{0}^{1} t^{\frac{n}{2}}(1-t)^{\frac{n-1}{2}}(x^{2}-t)^{-\frac{n+1}{2}}\, dt; \quad n > -1.

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Problem 5702

Let $\varphi (x,t) = t^{\beta -1}(1-t)^{\gamma -\beta -1}(1-tx)^{-\alpha }$ . Prove that each of the integrals

\int _{1}^{\infty } \varphi (x,t)\, dt; \qquad \alpha +1 > \gamma > \beta ;

\int _{0}^{1/x} \varphi (x,t)\, dt; \qquad \beta > 0, \ \alpha < 1;

\int _{0}^{1} \varphi (x,t)\, dt; \qquad \gamma > \beta > 0, \ \alpha < 1;

\int _{0}^{-\infty } \varphi (x,t)\, dt; \qquad \beta > 0, \ \alpha +1 > \gamma , \ \alpha < 1;

\int _{1}^{1/x} \varphi (x,t)\, dt; \qquad \gamma > \beta , \ \alpha < 1;

\int _{1/x}^{\infty } \varphi (x,t)\, dt; \qquad \alpha +1 > \gamma , \ \alpha < 1

is a particular integral of the hypergeometric equation

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

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Problem 5703

Show that the equation

(Ax^{2}+Bx+C)y''+(Dx+E)y'+Fy = 0

is reduced, by a simple change of variables, to the hypergeometric equation.

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Problem 5704

Prove the identity: $(1+x)^{n} = F(-n,\beta ,\beta ,-x)$ , where $F$ is the hypergeometric series (problem 5694).

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Problem 5705

Prove the identity: $e^{x} = \lim_{n\to \infty } F\left(1,n,1,\dfrac {x}{n}\right)$ , where $F$ is the hypergeometric series (problem 5694).

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Problem 5706

Prove the identity: $\ln (1+x) = xF(1,1,2,-x)$ , where $F$ is the hypergeometric series (problem 5694).

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Problem 5707

Prove the identity: $\dfrac {1}{2x}\ln \dfrac {1+x}{1-x} = F\left(\dfrac {1}{2},1,\dfrac {3}{2},x^{2}\right)$ , where $F$ is the hypergeometric series (problem 5694).

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Problem 5708

Prove the identity: $(1+x)^{n}+(1-x)^{n} = 2F\left(-\dfrac {n}{2},\dfrac {1-n}{2},\dfrac {1}{2},x^{2}\right)$ , where $F$ is the hypergeometric series (problem 5694).

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Problem 5709

Prove the identity: $\ch x = \lim_{\alpha ,\beta \to \infty } F\left(\alpha ,\beta ,\dfrac {1}{2},\dfrac {x^{2}}{4\alpha \beta }\right)$ , where $F$ is the hypergeometric series (problem 5694).

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Problem 5710

Prove the identity: $\cos n\arccos x = F\left(n,-n,\dfrac {1}{2},\dfrac {1-x}{2}\right)$ , where $F$ is the hypergeometric series (problem 5694).

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Problem 5711

Prove the identity:

J_{n}(x) = \lim _{\alpha ,\beta \to \infty } \left(\frac{x}{2}\right)^{n}\frac{1}{\Gamma (n+1)}F\left(\alpha ,\beta ,n+1,-\left(\frac{x}{2\sqrt{\alpha \beta }}\right)^{2}\right),

where $F$ is the hypergeometric series (problem 5694), and $J_{n}(x)$ is the Bessel function, equal to

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

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Problem 5712

Denoting $F\left(\alpha +n,\beta +n,\gamma +n,\dfrac {1-x}{2}\right)$ (problem 5694) by $F_{n}$ , prove that

\left((1-x)^{\gamma }(1+x)^{\alpha +\beta -\gamma +1}F_{1}\right)' = -2\gamma (1-x)^{\gamma -1}(1+x)^{\alpha +\beta -\gamma }F_{0}.

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Problem 5713

Using the preceding (problem 5712), show that

\left((1-x)^{\gamma +n-1}(1+x)^{\alpha +\beta -\gamma +n}F_{n}\right)^{(n)} = (-1)^{n}2^{n}\gamma (\gamma +1)\cdots (\gamma +n-1)(1-x)^{\gamma -1}(1+x)^{\alpha +\beta -\gamma }F_{0}.

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Problem 5714

Setting $\gamma = \beta +1$ , $\beta = -n$ , $\alpha = p+q+n+1$ for the Jacobi polynomial (problem 5581)

Q = \frac{n!}{(p+1)\cdots (p+n)}P_{n}^{(p,q)}(x) = F\left(p+q+n+1,-n,p+1,\frac{1-x}{2}\right),

prove the identity:

(1-x)^{p}(1+x)^{q}Q = \frac{(-1)^{n}}{2^{n}(p+1)\cdots (p+n)}\frac{d^{n}}{dx^{n}}\left((1-x)^{p+n}(1+x)^{q+n}\right).

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Problem 5715

In the particular case $p = q = 0$ (problem 5714), obtain the Rodrigues formula for the Legendre polynomials (problem 5582):

P_{n}(x) = \frac{(-1)^{n}}{2^{n}\cdot n!}\frac{d^{n}(1-x^{2})^{n}}{dx^{n}}.

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Problem 5716

Prove the identity for the Chebyshev polynomials (problem 5583):

T_{n}(x) = \frac{\cos n\arccos x}{2^{n-1}} = \frac{1}{2^{n-1}}F\left(n,-n,\frac{1}{2},\frac{1-x}{2}\right) = \frac{(-1)^{n}}{1\cdot 3\cdots (2n-1)}\cdot \frac{\sqrt{1-x^{2}}}{2^{n-1}}\cdot \frac{d^{n}(1-x)^{n-\frac{1}{2}}}{dx^{n}}.

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Problem 5717

Prove the identity: $\displaystyle \int_{0}^{x} u^{p-1}(1-u)^{q-1}\, du = \frac{x^{p}}{p}F(1-q,p,p+1,x)$ , where $F$ is the hypergeometric series (problem 5694).

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Problem 5718

Prove the identity: $\displaystyle \int_{0}^{x} u^{p-1}e^{-u}\, du = \frac{x^{p}}{p}\lim_{n\to \infty } F\left(1-n,p,p+1,\frac{x}{n}\right)$ , where $F$ is the hypergeometric series (problem 5694).

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Problem 5719

Prove the identity: $\displaystyle \int_{0}^{1} \frac{du}{\sqrt{(1-u^{2})(1-k^{2}u^{2})}} = \frac{\pi }{2}F\left(\frac{1}{2},\frac{1}{2},1,k^{2}\right)$ , where $F$ is the hypergeometric series (problem 5694).

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