The General Solution of a Differential Equation

The General Solution of a Differential Equation

Multiplicity of Solutions of a Differential Equation

Recall that when integrating, we get a constant of integration, i.e. integrating

\[y' = e^x\]

results in:

\[y = e^x + c\]

and integrating twice:

\[y'' = e^x\]

results in:

\[y = e^x + c_1x + c_2\]

For a large class of differential equations, the solution a diffrential equation of order $n$ has $n$ arbitrary constants. However, this isn’t always true.

Because all solutions to differential equations involve at least one arbitrary constant, any differential equation that has a solution has infinitely many solutions.

Definition 4.3 - n-parameter family of solutions

The functions defined by

\[y = f(x,c_1,c_2,\cdots,c_n) \tag{4.31}\]

of the $n + 1$ variables, $x, c_1, c_2, \cdots, c_n$ will be called an n-parameter family of solutions of the $n$th order differential equation

\[F(x,y,y',\cdots,y^{(n)} = 0 \tag{4.32}\]

if for each choice of a set of values $c_1, c_2, \cdots, c_n$, the resultion function $f(x)$ defined by (4.31) (it will now define a function of $x$ alone) satisfies (4.32), i.e. if

\[F(x,f',f'',\cdots,f^{(n)}) = 0 \tag{4.33}\]

For the classes of differential equations we shall consider, we can now assert: a differential equation of the *nth order has an n-parameter family of solutions*.

Finding a Differential Equation from its n-Parameter family of solutions

While an $n$-parameter family of solutions contains $n$ arbitrary constants, the differential equation it is a solution of will contain no constants.

The approach to find the differential equation, then, is to differentiate the $n$-parameter family of solutions and then eliminate any constants remaining.

For example, if we have the 1-parameter family of solutions

\[\tag{a} y = ccosx+x\]

we can differentiate it to get:

\[\tag{b} y' = -csinx + 1\]

We can now solve for $c$:

\[\frac{1 + y'}{sinx} = c\]

Then substitute $c$ into (a) to get:

\[y = \frac{1 + y'}{sinx}cosx+x\]

Simplifying we get:

\[y' = (x - y)tanx + 1,~x \neq \pm \frac{\pi}{2},\pm\frac{3\pi}{2},\ldots,\]

Which is the required differential equations. There are other ways to solve for $c$ - it’s just algebra at this point.

The same thing applies for $2$-parameter families of solutions - we’d just differentiate twice instead. This can, in some cases, such as $y =c_1e^x + c_2e^{-x}$ where we get two equations with two unknowns which can be simultaneously solved.

General and Particular Solutions

An $n$-parameter family of solutions is called a “general” solution in some sources, but they don’t always capture all solutions of a differential equation; there can also be particular solutions outside $n$-parameter family of solutions.

For example, the first order ODE:

\[y = xy' + (y')^2 \tag{4.6}\]

has for a solution the 1-parameter family

\[y = cx + c^2 \tag{4.61}\]

Some sources might call this a general solution, but it does not include all particular solutions, for example, the function

\[y = - \frac{x^2}{4} \tag{4.62}\]

is also a solution of 4.6 but is not included in 4.61.

In this text (Ordinary Differential Equations by Tenenbaum and Pollard) we’ll use these definitions:

Definition 4.66 - Particular Solutoin A solution of a differential equation will be called a particular solution if it satisfies the equation and does not contain arbitrary constants.

Definition 4.7 - General Solution A $n$-parameter family of solutions of a differential equation will be called a general solution if it contains every particular solution of the equation.

We won’t call a $n$-parameter family of solutions general unless we can prove it contains all particular solutions of the equation.

Definition 4.71 - Initial Conditions The $n$ conditions which enable us to determine the values of the arbitrary constants $c_1,c_2,\cdots,c_n$ in an $n$-parameter family, if given in terms of one value of the independent variable, are called initital conditions.

Normally the number of initial conditions must equal the order of the differential equation. There are exceptions, but for the ones considered in this book, this will be true.