Complete Integral, General Integral, Particular Solution, and Singular Solution

The complete general singular solution PDE structure helps us understand different solution types for a first-order partial differential equation . In this page, we study Complete Integral, General Integral, Particular Solution, and Singular Solution. These ideas are important because a first-order partial differential equation does not behave exactly like a first-order ordinary differential equation.

NOTETYPES OF SOLUTIONS

\begin{equation*} F(x,y,z,p,q)=0. \quad \text{(1)} \end{equation*}

A first-order PDE can have several related kinds of solutions. Choose a solution type and click Show Meaning to see its role in the solution structure. Notice that a complete integral uses two arbitrary parameters, while a general integral uses an arbitrary function. The particular and singular solutions are obtained in different ways, so separating these four ideas is important before using them in calculations.

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DEFINITIONCOMPLETE INTEGRAL

\begin{equation*} f(x,y,z,a,b)=0 \quad \text{(2)} \end{equation*}

The Complete Integral is useful because it can generate other types of solutions. It plays a role similar to a general solution in ordinary differential equations, but it is not the full general solution of a partial differential equation .

DEFINITIONGENERAL INTEGRAL

\begin{equation*} f(\phi,\psi)=0, \quad \text{(3)} \end{equation*}

a

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The main difference is important. A complete integral contains arbitrary constants. A general integral contains an arbitrary function.

DEFINITIONPARTICULAR SOLUTION

A particular solution is obtained when the arbitrary function or arbitrary parameters in a general form are chosen in a definite way.

a

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In applications, we usually need a particular solution. The reason is simple. Physical or geometrical conditions often select one solution from many possible solutions.

DEFINITIONSINGULAR SOLUTION

A singular solution is a solution obtained as the envelope of a two-parameter family of surfaces, when such an envelope exists.

The Singular Solution is special. It is not usually obtained by giving particular values to the arbitrary constants or arbitrary function. It comes from the envelope of the complete integral.

NOTEENVELOPE METHOD

\begin{equation*} f(x,y,z,a,b)=0. \quad \text{(4)} \end{equation*}

b=b(a)

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NOTESINGULAR SOLUTION BY ENVELOPE

\begin{equation*} f(x,y,z,a,b)=0, \qquad f_a(x,y,z,a,b)=0, \qquad f_b(x,y,z,a,b)=0. \quad \text{(7)} \end{equation*}

f=0

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REMARKDIFFERENCE FROM ORDINARY DIFFERENTIAL EQUATIONS

In a first-order ordinary differential equation, the general solution usually contains one arbitrary constant. In a first-order partial differential equation, the general integral contains an arbitrary function. This is why the solution set of a partial differential equation is much larger.

A first-order ODE and a first-order PDE have different kinds of solution freedom. Choose ODE, PDE, or both, then click Compare Freedom to see the difference. A first-order ODE usually has one arbitrary constant, while a first-order PDE general integral can contain an arbitrary function. This explains why the collection of PDE solutions is much larger.

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NOTESIMILAR TITLES

Some books use the phrase Complete Solution for Complete Integral. Some books also use Singular Integral for Singular Solution. In this page, we use Complete Integral and Singular Solution because these titles are clear for students.

FAQFrequently Asked Questions

5 Questions

A complete integral is a solution containing two arbitrary parameters.

A general integral is a solution containing an arbitrary function.

A particular solution is obtained after choosing the arbitrary function or parameters in a definite way.

A singular solution is obtained from the envelope of a two-parameter family, if that envelope exists.

No. The singular solution is usually obtained separately from the envelope of the complete integral.

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