First-order ordinary differential equations - separable equations, families of curves, and recovering an ODE from its general solution.
These practice problems are the entry to ordinary differential equations. The unknown is a function of one variable, and the equation involves that function and its first derivative. You classify the equation, solve it, and then impose an initial condition or a geometric side condition. Each solution is a full write-up of the method, not only the final family of curves.
Families of Curves asks you to recover a differential equation from a one-parameter family, and to go the other way. Separable and Homogeneous Equations is the first solving chapter. Linear Equations and Applications includes integrating factors and the standard models built from a linear equation. Exact Equations treats differentials that come from a potential. Equations Not Solved for the derivative, Trajectories, and Further Methods cover equations that become separable, linear, or exact after a substitution.
Begin with separable equations if the course is new, and use the difficulty control to keep the first pass short. Differential Calculus is the easier prerequisite if the derivative rules inside a solution are the obstacle. When first-order methods are familiar, continue with Second-Order Differential Equations, and later with Systems of Differential Equations.
Problem 1
MediumFamilies of Curves
Find the general solution of the differential equation whose solution is given by $y = C e^{\sqrt{x^{2}-1}}$.
Find the differential equation whose general solution is the one-parameter family $x^2+c\,y^2=2y$, and show that the family is recovered by solving that ODE.
By using a suitable substitution, bring the equation $(a^2+y^2)dx+2x\sqrt{ax-x^2}\,dy=0$ to a form with separated variables, and then integrate it, given $y(a)=0$.
By using a suitable substitution, bring the equation $x^2y'-\cos2x=1$ to a form with separated variables, and then integrate it, given $y(\infty)=\dfrac{9\pi}{4}$.
By using a suitable substitution, bring the equation $\dfrac{3y^2y'}{16x}=2xy^3$ to a form with separated variables and integrate it, given that $y(x)$ remains bounded as $x\to\infty$.
Write the equation of the curve that passes through $\left(1,\dfrac13\right)$ and for which the slope of the tangent line, at each point, is three times the slope of the radius vector at that point.
Find the curves for which the area of the triangle formed by the tangent line, the ordinate of the point of tangency, and the $x$-axis is constant and equal to $a^2$.
Find the family of curves for which the length of the tangent segment (from the point of tangency to the $x$-axis) equals the distance from the point of tangency to the origin.
For small quantities of radioactive material, the rate of decay is directly proportional to the amount of radioactive material present. Find the mass of radioactive material as a function of time.
A heavy point of mass $m$ begins to fall in a medium with resistance $f=kv^n$. Determine the velocity as a function of time and the distance fallen in time $t$, for $n=1$ and $n=2$.
A motorboat moving in still water at $v_0=20$ km/h has its motor turned off; 20 seconds later its velocity has dropped to $8$ km/h. Water resistance is proportional to velocity. Find the boat's velocity 20 minutes after the motor is turned off.
A tank holds 10 liters of water. Brine containing 0.3 kg of salt per liter flows in at 2 L/min, mixes instantly, and the mixture drains out at the same rate. How much salt is in the tank after 5 minutes?
What relation must hold between $\alpha$ and $\beta$ so that $y'=ax^\alpha+by^\beta$ can be reduced to a homogeneous equation by the substitution $y=z^m$?
Find the family of curves for which the length of the tangent (from the point of tangency to where it meets the $x$-axis) equals the segment the tangent cuts off on the positive $x$-axis.
Find the equation of the curve through $(1,0)$ for which the $y$-intercept of the tangent line equals the distance from the point of tangency to the origin.
A point $M$ on a curve $(c)$ projects onto the $y$-axis at $Q$, and the normal at $M$ meets the $x$-axis at $N$. Find the curves $(c)$ for which $OM\perp QN$, and the member through $(2,0)$.
Find the necessary and sufficient condition for the solutions of the homogeneous equation $y'=f\!\left(\dfrac{y}{x}\right)$ to be closed curves surrounding the origin.
Find the curves for which the tangent at each point forms with the coordinate axes and the ordinate of the point of tangency a trapezoid of constant area $a^2$. Among these curves, find the one through the point $(a,a)$.
If $R$ is the resistance, $L$ is the coefficient of self-inductance, and $U$ is the voltage of an electrical circuit, its dynamic equilibrium equation is $L\dfrac{di}{dt}+Ri=U$. Find the current as a function of time $t$ if $i=i_0$ when $t=0$.
A capacitor of capacitance $C$ is connected to a circuit with voltage $E$ and resistance $R$. Determine the charge $q$ on the capacitor at time $t$ after connection (uncharged at $t=0$).
In a circuit with resistance $R$ and self-inductance $L$, a periodic electromotive force $E_1=a\sin\dfrac{2\pi}{T}t$ acts. Determine the current $i$ at every moment if $i=0$ when $t=0$.
Find the family of curves for which at every point the sum of the length of the normal and the subnormal equals unity. Find the curve passing through $(0,0)$.
The tangent at an arbitrary point $M$ of a curve $c$ intersects the $y$-axis at point $N$. Determine the curves $c$ such that the segment $MN$ is seen from the point $(a,0)$ at a right angle.
Find the family of curves $c$ such that the tangent at an arbitrary point cuts off on the $y$-axis (or $x$-axis) an intercept: (1) proportional to the ordinate (or abscissa) of the point of tangency; (2) proportional to the third power of the ordinate (or abscissa) of the point of tangency.
Find the family of curves $r=f(\varphi)$ for which the area of the sector bounded by the curve and the radii from a fixed point $(r_0,\varphi_0)$ to a variable point $(r,\varphi)$ is proportional to the product $r\varphi$, with proportionality coefficient $k$.
For the Riccati equation $y'-\dfrac{1}{1-x^3}y^2+\dfrac{x^2}{1-x^3}y+\dfrac{2x}{1-x^3}=0$: (1) determine $a$ so that $y=ax^2$ is a particular integral and find the general solution; (2) determine $a,b$ so that $y=ax+b$ is a particular integral and find the general solution.
Determine constants $c,d$ such that $y=\dfrac{c}{x^2}+\dfrac{d}{x}$ is a particular integral of $y'+y^2-\dfrac{a^2}{x^4}=0$, and find the general integral.
Find the general integral of $\left(\dfrac1y\sin\dfrac{x}{y}-\dfrac{y}{x^2}\cos\dfrac{y}{x}+1\right)dx+\left(\dfrac1x\cos\dfrac{y}{x}-\dfrac{x}{y^2}\sin\dfrac{x}{y}+\dfrac{1}{y^2}\right)dy=0$.
From all integral curves of $\left(1+x\sqrt{x^2+y^2}\right)dx+\left(-1+\sqrt{x^2+y^2}\right)dy=0$, find the particular curve that passes through the point $(0,1)$.
Show that $\left(2xy+x^2y+\dfrac{y^3}{3}\right)dx+(x^2+y^2)dy=0$ has an integrating factor of the form $\mu=\mu(x)$ or $\mu=\mu(y)$, and find its general integral.
Show that $(x^2+y^2)(x\,dy-y\,dx)=(a+x)x^4\,dx$ (with parameter $a$) has an integrating factor of the form $\mu=\mu(x)$ or $\mu=\mu(y)$, and find its general integral.
The equation $(x\cos y-y\sin y)\,dx+(x\sin y+y\cos y)\,dy=0$ is claimed to have an integrating factor of the form $\mu=\mu(x)$ or $\mu=\mu(y)$. Test this claim rigorously and find the general integral if such a factor exists.
Show that $(2x+y-yx^2)\,dx+x(1+x^2)\,dy=0$ has an integrating factor and find its general integral. (The natural guess $\mu=\mu(x^2+y^2)$ is tested first; it turns out not to satisfy the required condition, but $\mu=\mu(x)$ does.)
Show that $(x^2+x^2y+2xy-y^2-y^3)\,dx+(y^2+xy^2+2xy-x^2-x^3)\,dy=0$ has an integrating factor of the form $\mu=\mu(x+y)$ or $\mu=\mu(xy)$, and solve it.
Solve $x=yy'+a(y')^2$ using the parameter method (roles of $x$ and $y$ swapped relative to standard Clairaut form), and determine the true singular solutions.
Find all curves $y=y(x)$ such that, for the tangent and normal drawn at an arbitrary point $M=(x,y)$ meeting the $x$-axis at $T$ and $N$ respectively, $OM^2=OT\cdot ON$ (O the origin).
Find the curve for which the product of the $y$-intercept of the tangent and the ordinate of the point of tangency equals $\tfrac{a}{2}$ times the excess of the length of the normal over the subnormal.
Find the singular solution (envelope) of the one-parameter family of curves $y^2=(x-c)^3$, and verify it against the differential equation of the family.
Find the orthogonal trajectories of the family $y^2=2p(x-a)$, where $p$ is a fixed given constant (the shared semi-latus-rectum) and $a$ is the family's varying parameter.
Find the orthogonal trajectories of the family of curves $r^{2n}-2a^nr^n\cos n\varphi+a^{2n}=c^{2n}$, where $a$ and $n$ are fixed given constants and $c$ is the family's varying parameter.
Find the isogonal trajectories, at constant angle $\alpha$, of the family of cardioids $r=a(1+\cos\varphi)$, and examine the orthogonal ($\alpha=90°$) limiting case.
Find the isogonal trajectories, at constant angle $\alpha$, of the family of lemniscates $r^2\cos2\varphi=a^2$, and examine the orthogonal ($\alpha=90°$) limiting case.
Show that \((x^2+x)y' + y^2 + (1-2x)y - 2x = 0\) has a particular solution of the form \(y_1=k\) for a constant \(k\), and show that all integral curves of the general solution pass through two fixed points.
Given the differential equation \(y'=f(x,y)\), suppose it admits an integrating factor that depends only on \(x\). Show that the equation is then linear.
Given the differential equation \((x^3 - 1)y' = 2xy^2 - x^2y - 1.\) \(1^\circ\) Determine a particular solution of the form \(y = ax + b\) and then find the general solution. \(2^\circ\) Find the particular solution \(y = \varphi(x)\) that satisfies \(y(0) = -\frac{1}{4}\).
Given the function \(y = \frac{x+1}{\sqrt{x^2+1}}.\) Show that it satisfies a differential equation of the form \(y'=R(x,y)\) where \(R\) is a rational function of \(x\) and \(y\). Then integrate this differential equation.
Show that the function \(y(x) = \begin{cases}\dfrac{x}{e^x-1}, & x \neq 0\\(4pt] 1, & x=0\end{cases}\) satisfies a first-order differential equation of the form \(y'=R(x,y)\), where \(R\) is a rational function in \(x\) and \(y\).
Show that the differential equation \(y' = \frac{(x^2+y^2-\alpha)y+\lambda x}{(x^2+y^2-k)x-\beta y}\) where \(\alpha,\beta,k,\lambda\) are parameters, can be reduced to separable form by the substitution \(x=\rho\cos\varphi\), \(y=\rho\sin\varphi\).
Determine the integral curves of the differential equation \(8xyy' - 16y^2 + x^2 = 0.\) Examine these curves and sketch them. If the integral curves bound a region of the plane, calculate the area of this region and the arc length of the curves.
Show that the differential equation \((y')^3-(y')^2-4y^2-27y^4=0\) can be represented by the relations \(y' = (1+t^2)(1+3t^2), \qquad y = t+t^3,\) where \(t\) is a parameter. Use this fact to find the solution of the given equation.
Given the differential equation \(y'+\dfrac{x}{y}=1\). \(1^\circ\) By substituting \(y=vx\), show that the solution is defined by \(\frac{\sqrt3}{2}\ln(x^2-xy+y^2)+\arctan\frac{2y-x}{x\sqrt3}=c_1.\) \(2^\circ\) By substituting \(x=vy\), show that the solution is defined by \(\frac{\sqrt3}{2}\ln(x^2-xy+y^2)-\arctan\frac{2x-y}{y\sqrt3}=c_2.\) \(3^\circ\) Show that these solutions are equivalent and find the relationship between \(c_1\) and \(c_2\).
Show that the function \(y(x)=\dfrac{\arcsin\sqrt x}{\sqrt{x-x^2}}\) is a solution of the differential equation \(y'=f(x)y+\varphi(x)\), where \(f(x)\) and \(\varphi(x)\) are rational functions of \(x\) to be determined. Show that a particular solution of this equation can be expanded in a power series in a neighborhood of \(x=0\).
Show that the integral curves of the equation \((4y-3x-5)y'-3y+7x+2=0\) form a family of closed curves. Show that the integral curves are symmetric with respect to the line \(y=1\).
\(1^{\circ}\) Show that there exists a function \(\mu=\mu(xy)\) such that the expression \(\mu(x^2y^2+x)\,dx + \mu y\,dy\) represents a total differential. \(2^{\circ}\) Determine the integral curves of the corresponding equation and identify the one passing through the point \((-1,1)\).
Given the differential equation \(\frac{1}{x}(\lambda x^2 - y^2)\,dx + (\mu x + 2\lambda y)\,dy = 0,\) \(1^{\circ}\) determine the parameters \(\lambda\) and \(\mu\) such that the equation has an integrating factor \(\dfrac{1}{x^2+y^2}\). \(2^{\circ}\) For the values of \(\lambda,\mu\) thus obtained, find the particular solution \(\varphi(x,y)=0\) satisfying \(\varphi(a,0)=0\) \((a>0)\).
Find the condition under which \(P\,dx+Q\,dy=0\) admits an integrating factor of the form \(\mu=\mu(y^2-x)\). Show that this factor can be obtained by quadrature. Apply this to the case \(P = (2y^2-2x-1)e^x+(2y^2-3x)e^y,\qquad Q = 2ye^x+2x(y^2+y-x)e^y.\)
Given \(P(x,y)\,dx+Q(x,y)\,dy=0\): \(1^{\circ}\) Find the relationship between \(P\) and \(Q\) such that the equation has an integrating factor \(\mu=\mu(x+y)\). \(2^{\circ}\) Apply this to \((2xy-y^2-y)\,dx+(2xy-x^2-x)\,dy=0.\) \(3^{\circ}\) When the equation simultaneously admits integrating factors \(\lambda(x+y)\) and \(\mu(x-y)\), express \(P\) and \(Q\) in terms of \(\lambda,\mu,\lambda',\mu'\).
Find the family of curves such that the tangent at an arbitrary point, the radius vector to that point, and the \(Ox\) axis form a triangle of constant area \(a\).
Find the curves \(y=f(x)\) such that the \(x\)-coordinate of the centroid of the curvilinear trapezoid bounded by the curve, the coordinate axes, and the line \(x=a\) equals \(\dfrac{3a}{4}\) for every \(a>0\).
Find the curves for which the sum of the normal and the subnormal is proportional to the abscissa: \(N+SN=kx\). For what value of the proportionality constant \(k\) is the resulting curve a genuine curve of the second degree?
In the plane, a point \(O\) and a line \(d\) are given. \(1^{\circ}\) Determine the curves \(c\) such that the distance from each point \(M\) of the curve to the line \(d\) equals the distance from \(O\) to the tangent to the curve at \(M\). \(2^{\circ}\) Through an arbitrary point \(A\) of the plane there generally pass two integral curves of the resulting differential equation. Find the locus of points \(A\) where the tangents of the two integral curves coincide.
Through an arbitrary point \(M\) of a curve \(c\) in the plane \(z=0\), a line \(MN\) is drawn such that the ratio of the slope of \(MN\) to the slope of the tangent to \(c\) at \(M\) is a constant real number \(a\). \(1^{\circ}\) Find the equation of curve \(c\) if the line \(MN\) is always tangent to the parabola \(y=\tfrac12x^2\). Distinguish the cases \(a=\tfrac12\) and \(a\neq\tfrac12\). \(2^{\circ}\) Find the equation of the ruled surface whose director plane is the coordinate plane \(z=0\) and whose generators are lines through the \(Oz\) axis and the curve \((L):\ x=\alpha(u),\,y=\beta(u),\,z=u\), if the projection of \((L)\) onto \(z=0\) is the curve \(c\) corresponding to \(a=\tfrac12\), where \(\mu\) is the parameter appearing in the equation of \(c\).
Find the locus of points such that the two integral curves of \(x(y')^2 - yy' + 1 = 0\) passing through each such point intersect: \(1^{\circ}\) at right angles; \(2^{\circ}\) at an angle of \(45^{\circ}\).
A particle falls in a medium whose resistance is proportional to the square of the particle's velocity. Show that the equation of motion is \(\frac{dv}{dt} = g - kv^2,\) where \(k\) is a constant and \(g\) is the acceleration due to gravity. Integrate this equation and show that \(v\to\sqrt{g/k}\) as \(t\to\infty\).
The velocity of a material point at any instant differs from the average velocity (from the beginning of motion to that instant) by a magnitude which is proportional to the kinetic energy of the point and inversely proportional to the time elapsed from the beginning. Find the dependence of distance on time.
A rocket of initial mass \(M_0\) moves in a straight line under the action of a force produced by the continuous ejection of a jet of gases expelled by the rocket. The ejection velocity \(u_0\) of the gases (relative to the rocket) is constant and oriented opposite to the initial velocity \(v_0\) of the rocket. Find the law of motion of the rocket, neglecting gravity and air resistance (Tsiolkovsky's problem).
At a distance \(a\) apart, at points \(A\) and \(B\), there are two oppositely charged electric charges \(+q\) and \(-q\). Taking point \(A\) as the coordinate origin and the line \(AB\) as the \(x\)-axis, formulate the equation of the equipotential lines of the electric field created by these charges.