Second-order ordinary differential equations - linear equations with constant coefficients, reduction of order, and initial-value problems.
These practice problems treat ordinary differential equations of the second order, mostly linear equations. You will solve the homogeneous equation from its characteristic polynomial, build a particular solution, reduce the order when one solution is known, and match initial values of the function and its first derivative. Solutions are written step by step, including the choice of undetermined coefficients or of a second independent solution.
Constant Coefficients is the core chapter: distinct real roots, complex roots, and repeated roots. Reduction of Order is for equations where one solution is already in hand and a second must be found. Nonhomogeneous Equations covers undetermined coefficients and variation of parameters. Initial-Value Problems asks for the single solution that matches both conditions at one point.
Find the integral curve of $y\,y'' + (y')^2 = 0$ which passes through the origin and whose tangent slope at any point equals the reciprocal of the ordinate at that point.
Find the curves for which $R = N^3,$ where $R$ is the radius of curvature and $N$ is the length of the normal (the segment of the normal between the point on the curve and the $x$-axis).
Find the curves for which $R = k\,r,$ where $R$ is the radius of curvature, $r=\sqrt{x^2+y^2}$ is the radius vector of the point, and $k>0$ is a constant.
Find the curves for which, at every point, $R = 2N,$ where $R$ is the radius of curvature and $N=y\sqrt{1+y'^2}$ is the length of the normal segment intercepted between the curve and the $x$-axis. Consider separately: $1^\circ$ the curve convex toward the $x$-axis ($y''>0$); $2^\circ$ concave toward the $x$-axis ($y''<0$).
Find the curves for which $R\cos\varphi = k \qquad (k>0\ \text{constant}),$ where $R$ is the radius of curvature and $\varphi$ is the angle the tangent makes with the $x$-axis ($\tan\varphi=y'$, $\cos\varphi=1/\sqrt{1+y'^2}$) — i.e. $R$ is inversely proportional to $\cos\varphi$.
Find the law of motion of a body sliding down a rough inclined plane, and the time required for it to slide down the entire length, given that it starts from rest at the top. Length $l = 10\text{ m}$, angle of inclination $\alpha = 45^\circ$, coefficient of friction $k = 0.5$, $g = 9.81\text{ m/s}^2$.
A cantilever beam is loaded by a force $F$ at its free end. Find the equation of the elastic curve and the deflection of the free end. Length $l = 6\text{ m}$, $F = 2\text{ t} = 2000\text{ kg}$, modulus of elasticity $E = 2{,}100{,}000\text{ kg/cm}^2$, moment of inertia $I = 30{,}000\text{ cm}^4$.
A simply supported beam of length $l$ carries a uniformly distributed load of total weight $P$. Find the equation of the elastic curve and the deflection at the midpoint.
Examine linear dependence of two general systems $1^\circ\quad 1,\ x,\ x^2,\ \ldots,\ x^n;\qquad\qquad 2^\circ\quad e^{k_1x},\ e^{k_2x},\ \ldots,\ e^{k_nx},$ where in $2^\circ$ the exponents are pairwise distinct: $k_i\neq k_j$ for $i\neq j$.
Find the homogeneous linear ODE of smallest order having the given functions as particular solutions $\begin{aligned} 1^\circ&\quad y_1=\sin x,\quad y_2=\cos x\\[2pt] 2^\circ&\quad y_1=e^x,\quad y_2=xe^x\\[2pt] 3^\circ&\quad y_1=x,\quad y_2=x^2\\[2pt] 4^\circ&\quad y_1=3x,\quad y_2=x-2,\quad y_3=e^x+1 \end{aligned}$. Method: the given functions must satisfy the Wronskian condition $W(y_1,\dots,y_n,y)=0$, which for elementary building blocks (exponentials, powers, trig) can be matched directly to a set of characteristic roots.
Find the homogeneous linear ODE of smallest order having the given functions as particular solutions $\begin{aligned} 1^\circ&\quad y_1=e^x,\quad y_2=\sinh x,\quad y_3=\cosh x\\[2pt] 2^\circ&\quad y_1=x,\quad y_2=x^3,\quad y_3=|x^3|\\[2pt] 3^\circ&\quad y_1=e^x,\quad y_2=e^x\sin x,\quad y_3=e^x\cos x \end{aligned}$. Method: the Wronskian condition $W(y_1,\dots,y_n,y)=0$, or — when the given set turns out to be dependent, as in $1^\circ$, or genuinely singular, as in $2^\circ$ — a preliminary reduction of the solution set before the ODE is formed.
Given the fundamental system $y_1=x$, $y_2=x^2$, $y_3=x^3$ of a homogeneous linear ODE, find the particular solution satisfying $y(1)=0$, $y'(1)=-1$, $y''(1)=2$.
Find the general solution of $x(x-1)\,y'' - (2x-1)\,y' + 2y = 2x^3-3x^2$, given that $y_1=x^2$ is a particular solution of the corresponding homogeneous equation.
Find the general solution of $x(x+1)\,y'' + (x+2)\,y' - y = x + \frac1x$, given that the corresponding homogeneous equation has a polynomial particular solution.
Find the general solution of $(x^2-1)\,y'' + 4x\,y' + 2y = 6x$, given that $y_1=x$ and $y_2=\dfrac{x^2+x+1}{x+1}$ are two particular solutions of this non-homogeneous equation.
Solve the boundary value problem ($a$, $l$, $b$ constant parameters) $y^{(4)} - a^4y = 0$, with $y(0)=0,\qquad y'(0)=0,\qquad y''(l)=0,\qquad y'''(l)=b$.
A point mass $m$ moves along the $x$-axis under an elastic (restoring) force $-kx$ and a friction force $-r\dot x$ proportional to velocity, where $k>0$ and $r\ge0$ are constants. Set up the equation of motion and classify the resulting behavior according to the strength of the friction. $m\ddot x + r\dot x + kx = 0$.
A material point of unit mass, $m=1$, moves along the $x$-axis under an elastic (restoring) force $-\omega^2x$ only, with no friction. Find the general law of motion, and the particular motion satisfying $x(0)=1$ (the initial velocity $v_0=\dot x(0)$ left free). $\ddot x + \omega^2x = 0$.
Find the law of motion of a physical pendulum: a rigid body of mass $m$ rotating freely about a fixed horizontal axis $O$ that does not pass through the center of mass $G$. Let $I$ be the moment of inertia about $O$, $l$ the distance from $O$ to $G$, $\varphi$ the angle of deflection from vertical equilibrium, and $g$ the acceleration due to gravity.
Find the law of motion of a point mass $m$ attracted toward a fixed center $O$ by a force whose magnitude is proportional to the distance $r$ from $O$: $F=-kr$, $k>0$.
Find the distance between two consecutive zeros of an arbitrary nontrivial solution of $y''+my=0, \qquad m=\mathrm{const}>0$. How many zeros can there be on an interval $(a,b)$?
Suppose $|f(t)|\lt c/t^{1+a}$ for $t\geq t_0$, with constants $c,a>0$. Prove that $u''-\bigl[1-f(t)\bigr]u=0$ has two solutions $u_1(t)=e^{t}\bigl[1+O(1/t^a)\bigr]$ and $u_2(t)=e^{-t}\bigl[1+O(1/t^a)\bigr]$ as $t\to\infty$.
Solve the initial value problem $y'' + \frac{1}{y^2}e^{y^2}y' - 2yy'^2 = 0, \qquad y\!\left(-\frac{1}{2e}\right)=1,\quad y'\!\left(-\frac{1}{2e}\right)=e$.
Find the general solution of $xyy'' + x(y')^2 = yy'$, then find the integral curve passing through both $(2,0)$ and $(0,1)$, and compute the area it encloses.
In the equation $y'' + 2y' + (\lambda^2+1)y = 0$, determine $\lambda$ so that some (nontrivial) solution vanishes at $x=0$ and at $x=2\pi$, with exactly two further zeros strictly between $0$ and $2\pi$.
Let $y_0(x)$ be a particular solution of the Bessel equation of order zero, $(E_0)\!: \qquad y'' + \frac1x y' + y = 0$, show that $y_0'(x)$ is a particular solution of the Bessel equation of order one, $(E_1)\!: \qquad y'' + \frac1x y' + \left(1-\frac{1}{x^2}\right)y = 0$.
Consider the two equations $(E_1)\!: \quad \sin x\,y'' - (\cos x+\sin x)y' + \cos x\,y = \cos x$ and $(E_2)\!: \quad y''' - y'' + y' - y = -1$. (a) Show that every solution of $(E_1)$ also satisfies $(E_2)$. (b) Solve both equations. (c) Find the periodic particular solution of $(E_1)$ satisfying $y(0)=2$.
For which value of $\alpha$ does the Legendre equation $(1-x^2)y'' - 2xy' + \alpha y = 0$ have a particular solution that is a degree-3 polynomial? Find that solution.
Show that $y = \frac{d^{\,n-1}}{dx^{\,n-1}}\Bigl[(1-x^2)^{\frac{2n-1}{2}}\Bigr]$ is a particular solution of $(x^2-1)y'' + xy' - n^2y = 0$, for $n\in\mathbb{N}$ and $|x|\lt 1$.
In the equation $(1+x^2)y'' - 2xy' + (ax^2+bx+c)y = 0$, determine the constants $a,b,c$ so that $y=e^{mx}$ is a particular solution, and find the general solution.
1° Find the general solution of the homogeneous equation $(H)\!: \qquad L(y) \equiv (2x+1)y'' + (4x-2)y' - 8y = 0$. 2° Find the general solution of the equation with the same left-hand side $(N)\!: \qquad (2x+1)y'' + (4x-2)y' - 8y = (2x+1)\sqrt{x}$.
Consider $(*)\!: \qquad y'' + \frac{4x-2}{2x+1}y' - \frac{8}{2x+1}y = 2(2x+1)^2e^{-x}$, whose homogeneous part has a particular solution of the form $y=e^{mx}$. 1° Find the general solution of $(*)$. 2° Find the integral curve tangent to the $x$-axis at the origin, i.e. satisfying $y(0)=0$, $y'(0)=0$.
Find the most general $u=u(r)$ satisfying Laplace's equation $\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0$ under the substitution $r=x^2+y^2$.
Find the most general $u=u(r)$ satisfying $\frac{\partial u}{\partial x}\frac{\partial^2u}{\partial y^2} - \frac{\partial u}{\partial y}\frac{\partial^2u}{\partial x^2} = 0$ under the substitution $r=y/x$.
Find the most general $u=u(r)$ satisfying $\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2} - \frac{4u}{x^2+y^2} = 0$ under the substitution $r=\sqrt{x^2+y^2}$.
Find the most general $u=u(r)$ satisfying the three-dimensional Laplace equation $\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}=0$ under the substitution $r=x^2+y^2+z^2$.
Find the most general $u=u(r)$ satisfying the three-dimensional Laplace equation $\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}=0$ under the substitution $r=\sqrt{x^2+y^2+z^2}$.
Solve the equation $(*)\!: \qquad y\,y'' + (y')^2 - \frac{y\,y'}{\sqrt{1+x^2}} = 0$: 1° by finding a multiplier that turns the left side into an exact derivative; 2° by the substitution $y'=yz(x)$. Show the two routes agree.
Given the equation $(*)\!: \qquad y'' = \alpha(x)y' + \beta(x)y, \qquad \alpha(x)\geq0,\ \beta(x)\geq0$, find the condition on $\alpha,\beta$ under which $(*)$ admits two linearly independent solutions $y_1,y_2$ with $y_1y_2=1$. Then, for $\alpha=1/x$, find $\beta$ and the general solution.
Find the condition between $f(x)$ and $\varphi(x)$ so that $(*)\!: \qquad y'' + f(x)y' + \varphi(x)y = 0$ has two solutions $y_1,y_2$ related by $y_2=y_1^2$.
Let $y = x^n e^{-x}$, where $n \in \mathbb{N}$. 1° Show that $x(y'+y) = ny$. 2° Show that $z = \dfrac{d^n}{dx^n}\bigl(x^n e^{-x}\bigr)$ satisfies $xz''+(x+1)z'+(n+1)z=0$. 3° Show that $w(x)=e^x z(x)$ is a polynomial of degree $n$ satisfying Laguerre's equation $xw''+(1-x)w'+nw=0$. Finally, evaluate $I = \displaystyle\int_0^\infty x^k\,\frac{d^n}{dx^n}\bigl(e^{-x}x^n\bigr)\,dx$ for $k\lt n$ and for $k=n$.
Let $y$ solve $y'' = f(x)\,y$, $(*)$, with $f$ differentiable. Show that $z=y^2$ satisfies a third-order linear homogeneous differential equation, and find it.
The equation $y''+P(x)y'+Q(x)y=0$, (1), is transformed by $x=\varphi(t)$ into $\ddot y+p(t)\dot y+q(t)y=0$, (2), where dots denote $d/dt$ and primes denote $d/dx$. Show that $\bigl[2P(x)Q(x)+Q'(x)\bigr]\,[Q(x)]^{-3/2} \;\equiv\; \bigl[2p(t)q(t)+\dot q(t)\bigr]\,[q(t)]^{-3/2}$.
Consider the Emden equation $y'' + \dfrac{2}{x}y' + y^\alpha = 0$, (E), with $\alpha\in\mathbb{R}$. 1° Show that if $f(x)$ solves (E), then so does $g(x)=A^{2/(\alpha-1)}f(Ax)$, for any $A\gt 0$. 2° For $\alpha=5$, find a particular solution of $y''+\dfrac{2}{x}y'+y^5=0$.
Show that $z(x) = \frac{d^n}{dx^n}\Bigl[(x-1)^{n+p}(x+1)^{n+q}\Bigr]$ is a particular solution of the Jacobi-type equation $(E)\!: \qquad (1-x^2)y'' + \bigl[(p+q-2)x + (p-q)\bigr]y' + (n+1)(n+p+q)\,y = 0$.
Show that the substitution $x=\varepsilon e^t$, $y=\varepsilon e^t\varphi(t)$ ($\varepsilon=\pm1$) reduces $(*)\!: \qquad x^2y'' - y(y')^2 + y = 0$ to the first-order equation $\dfrac{dz}{d\varphi} = \dfrac{\varphi(z^2-1)}{z-\varphi}$, where $z = \dfrac{d\varphi}{dt}+\varphi$.
1° Show that $(*)\!: \qquad xy''' + 4x^2y'' + 4x(2-x^2)y' - 12x^2y - 20x^4 = 0$ is the derivative of a second-order equation. 2° Find the general solution.
Consider the equation $(*)\!: \qquad \frac{d}{dx}\Bigl[(1-x^2)\frac{dy}{dx}\Bigr] + ay = 0, \qquad \text{i.e.} \qquad (1-x^2)y''-2xy'+ay=0$. 1° Show that if $y=f_1+f_2$ ($f_1$ even, $f_2$ odd) solves $(*)$, then $f_1$ and $f_2$ individually solve $(*)$. 2° Find $a$ so that $(*)$ has a degree-3 polynomial solution; give that solution and the general solution.
Consider $y'' + k^2 y = f(x), \qquad k \in \mathbb{R},\ k \neq 0$. Show that the general solution can be written as $ \displaystyle y = A\cos(kx) + B\sin(kx) + \frac{1}{k}\int_{x_0}^{x} f(t)\sin\bigl(k(x-t)\bigr)\,dt $, where $A, B$ are integration constants and $x_0$ is a fixed reference point.
Consider $y(x) = \int_0^1 (1-t^2)^{\lambda}\cos(xt)\,dt, \qquad \lambda > -1$. Show that $y(x)$ is a particular solution of $ x y'' + 2(\lambda+1)y' + xy = 0 $, and that for $\lambda = n \in \mathbb{N}$ the integral reduces to an elementary closed form.
Consider $(1+x^2)y'' + xy' - \tfrac{1}{4}y = 0$. Show that $y_1 = \sqrt{\dfrac{1+ix}{2}}$ and $y_2 = \sqrt{\dfrac{1-ix}{2}}$ are particular solutions, and deduce that $y = \sqrt{1+\sqrt{1+x^2}}$ is also a particular solution.
Consider $y'' + ay' + by = f(x), \qquad \lambda := 4b - a^2 > 0$. Show that the general solution can be written $y = Ae^{-\frac{a}{2}x}\sin\!\Big(\tfrac{\sqrt\lambda}{2}(x-B)\Big) + \frac{2}{\sqrt\lambda}\int_{x_0}^{x} f(t)\,e^{\frac{a}{2}(t-x)}\sin\!\Big(\tfrac{\sqrt\lambda}{2}(x-t)\Big)\,dt$.
Consider $y'' = f(x), \qquad y(a)=0,\ \ y(b)=0 \qquad (a\ne b)$. Show that the solution of this two-point boundary-value problem is $y = \frac{x-a}{b-a}\int_x^b (t-b)f(t)\,dt + \frac{x-b}{b-a}\int_a^x (t-a)f(t)\,dt$.
Consider $y'' + P(x)y' + Q(x)y = 0$. 1° Find the condition on $P(x)$ and $Q(x)$ under which a change of independent variable $t=f(x)$ reduces this equation to one with constant coefficients, $\ddot y + a\dot y + by = 0$. 2° Apply this to $P(x)=2/x$: find the corresponding $Q(x)$, and solve the resulting constant-coefficient equation.
Consider $(x^2-1)y'' + \bigl[x-(\alpha+\beta)\sqrt{x^2-1}\bigr]y' + \alpha\beta y = 0, \qquad x>1$. 1° Using the substitution $x=\cosh t$ ($t>0$), reduce the equation to one with constant coefficients and write down its general solution. 2° For $\alpha=5$, $\beta=3$, find the solution that behaves like $x^3$ as $x\to\infty$. 3° Determine the conditions on $\alpha,\beta$ under which every solution stays bounded as $x\to\infty$.
Consider $R=\frac{(1+y'^2)^{3/2}}{|y''|}, \qquad N = y\sqrt{1+y'^2}, \qquad R=\lambda N+\mu$. 1° Find the curves for which the radius of curvature equals twice the length of the normal, $R=2N$. 2° More generally, find the curves satisfying the linear relation $R=\lambda N+\mu$ between the radius of curvature and the normal.
Consider $A=\Bigl(x-\frac{y}{y'},\,0\Bigr), \qquad C=\Bigl(x-\frac{y'(1+y'^2)}{y''},\; y+\frac{1+y'^2}{y''}\Bigr)$. At each point $M=(x,y)$ of a curve $y=y(x)$, let $A$ be the point where the tangent meets the $x$-axis, and let $C$ be the center of curvature. Find the curves for which the $x$-projection of $C$ always equals the $x$-coordinate of $A$. In particular, find the curve tangent to the $y$-axis at $(0,a)$.
Consider $\mathbf r(x) = (x,\,y(x),\,x^2)$. A curve lies on the surface $z=x^2$, parametrized by $x$. Let the osculating plane at each point meet the $x$-axis at $A$ and the $y$-axis at $B$, and let $K$ be the midpoint of $AB$. Find the curves for which the tangent line at every point passes through $K$.
Find the curves for which the radius of curvature at every point equals the length of the segment of the normal line intercepted between the point on the curve at ordinate $y=h$ and the $x$-axis $(y=0)$, then find the particular curve satisfying $y=0$ at $x=-\tfrac{\pi h}{4}$ and $y'=1$ at $x=0$, and determine $\lim\limits_{y\to\infty} x$.
A particle of mass $m$ and density $\rho$ sinks slowly, starting from rest, in a viscous fluid of density $\rho_0$. The resistance of the fluid is proportional to the velocity, with proportionality constant $k$. Find the law of motion $x(t)$, where $x$ is the depth of the particle at time $t$.
A body falls from rest under gravity through a resisting medium whose resistance is proportional to the square of the velocity. Given that the terminal velocity is $v_\infty = 75\ \text{m/s}$ and $g=9.81\ \text{m/s}^2$, find the velocity $v(t)$ and the distance fallen $x(t)$ as functions of time.