Linear Equations with Constant Coefficients and Equations Reducible to Them

Write the general integral of the equation $y''-y=0$.

Write the general integral of the equation $y''-5y'+6y=0$.

Write the general integral of the equation $y''+3y'+2y=0$.

Write the general integral of the equation $y''+2y'+y=0$.

Write the general integral of the equation $y''-4y'+4y=0$.

Write the general integral of the equation $y''+2y'+5y=0$.

Write the general integral of the equation $y''+4y'+13y=0$.

Write the general integral of the equation $y''+y'+y=0$.

Write the general integral of the equation $y''+y=0$.

Write the general integral of the equation $y'''-8y=0$.

Write the general integral of the equation $y^{IV}-16y=0$.

Write the general integral of the equation $y^{IV}+4y=0$.

Write the general integral of the equation $y'''-6y''+11y'-6y=0$.

Write the general integral of the equation $y'''-5y''+17y'-13y=0$.

Write the general integral of the equation $y^{IV}-5y''+4y=0$.

Write the general integral of the equation $y^{IV}+5y''+4y=0$.

Write the general integral of the equation $y'''-6y''+12y'-8y=0$.

Write the general integral of the equation $y^{IV}+2y'''+3y''+2y'+y=0$.

Write the general integral of the equation $y^{IV}+4y'''+6y''+4y'+y=0$.

Write the general integral of the equation $y^{IV}-y'''+y''-y'+12y=0$.

Write the general integral of the equation $y^{VII}+3y^{VI}+3y^{V}+3y^{IV}=0$.

Find the particular integral of the equation $y^{IV}+a^{4}y=0$ for $x=0$, $y=0$, $y'=0$, $y''=0$, $y'''=1$.

Find the particular integral of the equation $y^{IV}+4y=0$ for $x=0$, $y=0$, $y'=0$; for $x=\pi$, $y=0$, $y'=a$.

Find the particular integral of the equation $y^{IV}+4y=0$ for $x=0$, $y=0$, $y'=1$; for $x=+\infty$, $y=0$.

Find the particular integral of the equation $y^{IV}-a^{4}y=0$ for $x=0$, $y=0$, $y'=0$; for $x=l$, $y''=0$, $y'''=b$.

For which values of $a$ will there exist solutions of the previous problem that are not identically zero, in the case $b=0$?

Find the general integral of the equation $y''+a^{2}y=e^{x}$ by the method of undetermined coefficients.

Find the general integral of the equation $y''-y=xe^{x}+e^{2x}$ by the method of undetermined coefficients.

Find the general integral of the equation $y''-3y'+2y=e^{3x}(x^{2}+x)$ by the method of undetermined coefficients.

Find the general integral of the equation $y'''-4y'=xe^{2x}+\sin x+x^{2}$ by the method of undetermined coefficients.

Find the general integral of the equation $y''+2y'+y=e^{-x}\cos x+xe^{-x}$ by the method of undetermined coefficients.

Find the general integral of the equation $y''+2y'+2y=e^{-x}\cos x+xe^{-x}$ by the method of undetermined coefficients.

Find the general integral of the equation $y''-7y'+6y=\sin x$ by the method of undetermined coefficients.

Find the general integral of the equation $y''-2y'+y=\sin x+e^{-x}+e^{x}$ by the method of undetermined coefficients.

Find the general integral of the equation $y''+y=2\sin x\sin 2x$ by the method of undetermined coefficients.

Find the general integral of the equation $y''-a^{2}y=e^{bx}$ for $b\neq a$ and $b=a$, by the method of undetermined coefficients.

Find the general integral of the equation $y''+a^{2}y=\sin bx$ for $b\neq a$ and $b=a$, by the method of undetermined coefficients.

Find the general integral of the equation $y^{IV}+2y'''+5y''+8y'+4y=\cos x+40e^{x}$ by the method of undetermined coefficients.

Find the general integral of the equation $y''+4y'+20y=a\sin 4x$ by the method of undetermined coefficients.

Find the general integral of the equation $y'''-2y'+4y=e^{-x}\cos x+e^{x}\sin x$ by the method of undetermined coefficients.

Find the general integral of the equation $y'''+y''-y'+15y=\sin 2x$ by the method of undetermined coefficients.

Find the general integral of the equation $y''+n^{2}y=\sin^{3}nx$ by the method of undetermined coefficients.

Find the general integral of the equation $y''+n^{2}y=ax\sin mx+b\cos mx$; $m\neq n$, by the method of undetermined coefficients.

Find the general integral of the equation $y''+n^{2}y=ax\sin nx+b\cos nx$ by the method of undetermined coefficients.

Solve the equation $y''+y=\tg x$.

Solve the equation $y''-y=\dfrac {2e^{x}}{e^{x}-1}$.

Solve the equation $y''+y=\dfrac {1}{\cos x}$.

Solve the equation $y''+y=\dfrac {1}{\cos 2x\sqrt{\cos 2x}}$.

Solve the equation $y''-6y'+9y=\dfrac {9x^{2}+6x+2}{x^{3}}$.

Solve the equation $y''-6y'+9y=36\sqrt{x}-\dfrac {12x+1}{x\sqrt{x}}$.

Integrate the equation $x^{2}y''+xy'+y=0$.

Integrate the equation $x^{2}y''+xy'-y=0$.

Integrate the equation $x^{2}y'''=2y'$.

Integrate the equation $x^{3}y'''+xy'-y=0$.

Integrate the equation $x^{3}y'''+2x^{2}y''-xy'+y=0$.

Integrate the equation $x^{2}y''+xy'+y=x$.

Integrate the equation $x^{2}y''-xy'+y=2x$.

Integrate the equation $x^{2}y''-2xy'+2y=2x^{3}-x$.

Integrate the equation $x^{3}y'''-x^{2}y''+2xy'-2y=x^{3}+3x$.

Integrate the equation $(3x+2)^{2}y''+7(3x+2)y'=-63x+18$.

Integrate the equation $x^{2}y''-4xy'+6y=2ax+\dfrac {12b}{x}$.

Integrate the equation $x^{4}y^{IV}+6x^{3}y'''+5x^{2}y''-xy'+y=x^{2}$.

Integrate the equation $(x+1)^{3}y''+3(x+1)^{2}y'+(x+1)y=6\ln (x+1)$.

Integrate the equation $x^{3}y''-x^{2}y'-3xy+16\ln x=0$.

Integrate the equation $(x+1)^{2}y''+(x+1)y'+y=x^{2}+2\sin \ln (1+x)$.

Integrate the equation $x^{2}y''+xy'-y=\dfrac {x}{(x+1)^{2}}-\ln (x+1)$.

Integrate the equation $x^{4}y'''+2x^{3}y''=1$ under the condition: if $x=1$, then $y=0$, $y'=0$, $y''=0$.

Find the particular integral of the equation $y''-(\alpha +\beta )y'+\alpha \beta y=Ae^{\gamma x}$ ($\alpha \neq \beta \neq \gamma$), satisfying the supplementary conditions: at $x=0$, $y=a$, $y'=b$.

Find the particular integral of the equation $y''-(\alpha +\beta )y'+\alpha \beta y=Ae^{\alpha x}$ ($\alpha \neq \beta$), satisfying the supplementary conditions: at $x=0$, $y=a$, $y'=b$. Obtain the required solution:

Find the particular integral of the equation $y''-2\alpha y'+\alpha^{2}y=Ae^{\alpha x}$, satisfying the supplementary conditions: at $x=0$, $y=a$, $y'=b$. Obtain the required solution:

Find the particular integral of the equation $y'''-5y''+8y'-4y=e^{2x}+e^{3x}$, satisfying the supplementary conditions: at $x=0$, $y=1$, $y'=0$, $y''=0$.

Find the particular integral of the equation $y''+n^{2}y=h\sin^{2}nx$, satisfying the supplementary conditions: at $x=0$, $y=0$, $y'=0$.

Find the particular integral of the equation $y''+4n^{2}y=h\sin^{2}nx$, satisfying the supplementary conditions: at $x=0$, $y=a$, $y'=0$.

Find the particular integral of the equation $y'''+y=1$, satisfying the supplementary conditions: at $x=0$, $y=0$, $y'=0$, $y''=0$.

Find the particular integral of the equation $y^{IV}+y=1$, satisfying the supplementary conditions: at $x=0$, $y=0$; at $x=\infty$, $y$ is finite.

Find the integral of the equation $y^{IV}-2y'''+2y''-2y'+y=\dfrac {\pi }{2}+4\cos x$, satisfying the conditions $y=y'=0$ at $x=0$ and at $x=\pi$.

Find a function of $x$, continuous with continuous derivative for $0<x<2$, satisfying the equation $y''-y=0$ for $x<1$ and the equation $y''-4y=0$ for $x>1$, and such that $y=0$ and $y'=1$ at $x=0$.

A body falls from height $h$ under the action of gravity $f_{1}=-mg$; friction forces, proportional to the velocity, act on it: $f_{2}=-kv$. The initial velocity is zero. Find the formula for the height $y$ after $t$ seconds from the start of the motion.

A body of mass $m$ slides along a plane inclined at angle $\alpha$ under the action of gravity. The friction force is given by the formula $f=-lp-kv$, where $p$ is the normal pressure, $v$ is the velocity of the body, $l$ and $k$ are constants. Find the dependence of the distance traveled on time.

A body of mass $m$ is suspended on a spring and is in equilibrium. The spring is then stretched by a length $h$ by a force $kh$ and the body is released. Determine the subsequent motion of the body, assuming the resistance of the medium is proportional to the velocity $f=-nv$.

By Newton's law, for the motion of a body along the axis $Ox$, $m\dfrac {d^{2}x}{dt^{2}}=f$, where $f$ is the force acting on the body. Find the motion of a body acted on simultaneously by two forces:

1. an elastic force, proportional to the displacement from the equilibrium position: $f_{1}=-a^{2}x$;

2. a friction force, proportional to the velocity: $f_{2}=-k\dfrac {dx}{dt}$.

Using the same notation as in problem 4368, find the motion of a body of mass $m$ acted on by two forces: $f_{1}=-a^{2}x$ and $f_{2}=b\sin nt$. Consider the cases $n\neq \dfrac {a}{\sqrt{m}}$ and $n=\dfrac {a}{\sqrt{m}}$. At $t=0$ the quantities $x$ and $\dfrac {dx}{dt}$ equal $0$.

A body of mass $m$ slides along a horizontal plane owing to a push that imparted an initial velocity $v_{0}$. A friction force equal to $-km$ acts on the body. Find the distance the body is able to travel.

Two forces act on a body of unit mass: $f_{1}=-k^{2}x$ and $f_{2}=-fk^{2}\operatorname {sign}\left(x'\right)$, equal to $fk^{2}$ and directed opposite to the velocity. The initial velocity is $x'=0$, and the initial distance from the origin is $x_{0}=nf$. Find the motion of the body.

A string of length $l$ is located along the axis $Ox$ between the points $(0,0)$ and $(l,0)$ and vibrates in the plane $xOy$. The displacement $y$ at the point $x$ and at time $t$ satisfies the equation $\dfrac {\partial^{2}y}{\partial t^{2}}=a^{2}\dfrac {\partial^{2}y}{\partial x^{2}}$. Find the types of vibration for which $y=u(x)\sin bt$, where $b$ characterizes the frequency of vibration. The ends of the string are fixed: $y=0$ at $x=0$ and $x=l$ for any $t$.

In the steady-state distribution of heat in a homogeneous body, the temperature $u$ satisfies Laplace's equation $\dfrac {\partial^{2}u}{\partial x^{2}}+\dfrac {\partial^{2}u}{\partial y^{2}}+\dfrac {\partial^{2}u}{\partial z^{2}}=0$. The interior of a cylindrical tube with homogeneous walls is filled with a liquid at temperature $t_{1}$, and on the outside with a liquid at temperature $t_{2}$. Find the distribution of temperature in the walls.

The same question as in the previous problem, for the layer between concentric spheres of radii $r_{1}$ and $r_{2}$.

A beam is located along the axis $Ox$ from the point $(0,0)$ to the point $(l,0)$. Its deflection at the point $x$ is equal to $y$ and satisfies the equation $y^{IV}=af(x)$, where $f(x)$ is the load, and $a$ is a coefficient depending on the material of the beam and the shape of its cross-section. Find the deflection of the beam for $f(x)=1$, if the beam is fixed at the ends ($y=y'=0$ at $x=0$ and $x=l$).

A rod between the same points as in the previous problem rotates about the axis $Ox$. Its deflection $y$ from the axis $Ox$, if possible, satisfies the equation $y^{IV}-a^{4}y=0$, where $a$ depends on the speed and properties of the rod. At $x=0$ and $x=l$ the quantities $y$ and $y''$ are equal to zero. For which $a$ is a deflection possible?

An elastic thin plate lies in the plane $xOy$ and is bent by distributed forces of intensity $p(x,y)$ acting on it. The deflection $w$ at the point $(x,y)$ (displacement from the plane $xOy$) satisfies the differential equation

(the constant $D$ is the rigidity of the plate).

Determine the deflection $w(r)$ for a circular, uniformly loaded plate of radius $a$, with a clamped edge. At the clamped edge, $w=0$ and $\dfrac {dw}{dr}=0$; at the center of the plate, $w$ and $\dfrac {d^{2}w}{dr^{2}}$ are finite ($r=\sqrt{x^{2}+y^{2}}$).

Solve the previous problem for a uniformly loaded circular plate of radius $a$, simply supported at the edge. At the supported edge $r=a$ the boundary conditions have the form: $w=0$ and $\dfrac {d^{2}w}{dr^{2}}+\dfrac {\nu }{r}\dfrac {dw}{dr}=0$ (the constant $\nu$ is Poisson's ratio); at the center of the plate the conditions are as before.