A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
, and other sciences. The following examples show how to solve differential equations in a few simple cases when an exact solution exists.
Equations in the form are called separable and solved by and thus . Prior to dividing by , one needs to check if there are stationary (also called equilibrium)
solutions satisfying .
Separable (homogeneous) first-order linear ordinary differential equations
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
of the first order
must be homogeneous and has the general form
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....
(moving the y terms to one side and the t terms to the other side),
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation....
in this case involves dividing by y, we must check if the constant function y=0 is a solution of the original equation. Trivially, if y=0 then y'=0, so y=0 is actually a solution of the original equation. We note that y=0 is not allowed in the transformed equation.
We solve the transformed equation with the variables already separated by Integrating,
where C is an arbitrary constant. Then, by exponentiation, we obtain .
Here, , so . But we have independently checked that y=0 is also a solution of the original equation, thus.
with an arbitrary constant A, which covers all the cases. It is easy to confirm that this is a solution by plugging it into the original differential equation:
Some elaboration is needed because ƒ(t) might not even be integrable. One must also assume something about the domains of the functions involved before the equation is fully defined. The solution above assumes the real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
case.
If is a constant, the solution is particularly simple, and describes, e.g., if , the exponential decay of radioactive material at the macroscopic level. If the value of is not known a priori, it can be determined from two measurements of the solution. For example,
gives and .
Non-separable (non-homogeneous) first-order linear ordinary differential equations
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s) are not separable. They can be solved by the following approach, known as an integrating factor
Integrating factor
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus, in this case often multiplying through by an...
method. Consider first-order linear ODEs of the general form:
The method for solving this equation relies on a special integrating factor, μ:
We choose this integrating factor because it has the special property that its derivative is itself times the function we are integrating, that is:
Multiply both sides of the original differential equation by μ to get:
Because of the special μ we picked, we may substitute dμ/dx for μp(x), simplifying the equation to:
Using the product rule in reverse, we get:
Integrating both sides:
Finally, to solve for y we divide both sides by :
Since μ is a function of x, we cannot simplify any further directly.
A simple example
Suppose a mass is attached to a spring which exerts an attractive force on the mass proportional
Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...
to the extension/compression of the spring. For now, we may ignore any other forces (gravity, friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
, etc.). We shall write the extension of the spring at a time t as x(t). Now, using Newton's second law
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
we can write (using convenient units):
where m is the mass and k is the spring constant that represents a measure of spring stiffness. Let us for simplicity take m=k as an example.
If we look for solutions that have the form , where C is a constant, we discover the relationship , and thus must be one of the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s or . Thus, using Euler's theorem we can say that the solution must be of the form:
To determine the unknown constants A and B, we need initial conditions, i.e. equalities that specify the state of the system at a given time (usually t = 0).
For example, if we suppose at t = 0 the extension is a unit distance (x = 1), and the particle is not moving (dx/dt = 0). We have
Simple harmonic motion can serve as a mathematical model of a variety of motions, such as the oscillation of a spring. Additionally, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum and molecular vibration....
The above model of an oscillating mass on a spring is plausible but not very realistic: in practice, friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
will tend to decelerate the mass and have magnitude proportional to its velocity (i.e. dx/dt). Our new differential equation, expressing the balancing of the acceleration and the forces, is
where is the damping coefficient representing friction. Again looking for solutions of the form , we find that
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
which we can solve. If there are two complex conjugate roots a ± ib, and the solution (with the above boundary conditions) will look like this:
Let us for simplicity take , then and .
The equation can be also solved in MATLAB symbolic toolbox as
x = dsolve('D2x+c*Dx+k*x=0','x(0)=1','Dx(0)=0')
although the solution looks rather ugly,
x = (c + (c^2 - 4*k)^(1/2))/(2*exp(t*(c/2 - (c^2 - 4*k)^(1/2)/2))*(c^2 - 4*k)^(1/2)) -
(c - (c^2 - 4*k)^(1/2))/(2*exp(t*(c/2 + (c^2 - 4*k)^(1/2)/2))*(c^2 - 4*k)^(1/2))
In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...
. The plot of displacement against time would look like this:
which does resemble how one would expect a vibrating spring to behave as friction removed the energy from the system.
To generate such a picture with MATLAB symbolic toolbox, run
x = dsolve('D2x+c*Dx+k*x=0','x(0)=1','Dx(0)=0')
a=sym('a'); b=sym('b'); cs=-2*a; ks=a.^2+b.^2;
a=-0.25; b=1; c=subs(cs); k=subs(ks); x1=subs(x);
a=-0.5; b=1; c=subs(cs); k=subs(ks); x2=subs(x);
a=-0.5; b=.5; c=subs(cs); k=subs(ks); x3=subs(x);
h3 = ezplot(x3, [0 10]); set(h3, 'Color', 'r'); hold on;
h2 = ezplot(x2, [0 10]); set(h2, 'Color', 'g'); grid on;
h1 = ezplot(x1, [0 10]); set(h1, 'Color', 'b'); title('damped oscillation');
legend([h1 h2 h3],'a=-0.25; b=1;','a=-0.5; b=1;','a=-0.5; b=.5;')
Linear systems of ODEs
The following example of a first order linear systems of ODEs
can be easily symbolically solved
in WolframAlpha.
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
In mathematics, an ordinary differential equation of the formis called a Bernoulli equation when n≠1, 0, which is named after Jakob Bernoulli, who discussed it in 1695...
are called separable and solved by 
and thus
. Prior to dividing by 
, one needs to check if there are stationary (also called equilibrium)
satisfying 
.
is some known function
.
, so 
. But we have independently checked that y=0 is also a solution of the original equation, thus
.
is a constant, the solution is particularly simple, 
and describes, e.g., if 
, the exponential decay of radioactive material at the macroscopic level. If the value of 
is not known a priori, it can be determined from two measurements of the solution. For example,
and 
.
:
, where C is a constant, we discover the relationship 
, and thus 
must be one of the complex number
or 
. Thus, using Euler's theorem we can say that the solution must be of the form:
is the damping coefficient representing friction. Again looking for solutions of the form 
, we find that
there are two complex conjugate roots a ± ib, and the solution (with the above boundary conditions) will look like this:
, then 
and 
.
