# Differential Equations¶

In this chapter, some facilities for solving differential equations are described. Currently only simple equations without auxiliary conditions are supported.

OdeSolve(expr1==expr2)

general ODE solver

Param expr1,expr2

expressions containing a function to solve for

This function currently can solve second order homogeneous linear equations with real constant coefficient. The solution is returned with unique constants generated by {UniqueConstant}. The roots of the auxiliary equation are used as the arguments of exponentials. If the roots are complex conjugate pairs, then the solution returned is in the form of exponentials, sines and cosines. First and second derivatives are entered as y', y''. Higher order derivatives may be entered as y(n), where n is any positive integer.

Example

In> OdeSolve( y'' + y == 0 )
Out> C42*Sin(x)+C43*Cos(x);
In> OdeSolve( 2*y'' + 3*y' + 5*y == 0 )
Out> Exp(((-3)*x)/4)*(C78*Sin(Sqrt(31/16)*x)+C79*Cos(Sqrt(31/16)*x));
In> OdeSolve( y'' - 4*y == 0 )
Out> C132*Exp((-2)*x)+C136*Exp(2*x);
In> OdeSolve( y'' +2*y' + y == 0 )
Out> (C183+C184*x)*Exp(-x);

Solve(), RootsWithMultiples()

OdeTest(eqn, testsol)

test the solution of an ODE

Param eqn

equation to test

Param testsol

test solution

This function automates the verification of the solution of an ODE. It can also be used to quickly see how a particular equation operates on a function.

Example

In> OdeTest(y''+y,Sin(x)+Cos(x))
Out> 0;
In> OdeTest(y''+2*y,Sin(x)+Cos(x))
Out> Sin(x)+Cos(x);

OdeSolve()

OdeOrder(eqn)

return order of an ODE

Param eqn

equation

This function returns the order of the differential equation, which is order of the highest derivative. If no derivatives appear, zero is returned.

Example

In> OdeOrder(y'' + 2*y' == 0)
Out> 2;
In> OdeOrder(Sin(x)*y(5) + 2*y' == 0)
Out> 5;
In> OdeOrder(2*y + Sin(y) == 0)
Out> 0;

OdeSolve()

WronskianMatrix(func, var)

create the Wronskian matrix

Param func

an $$n$$-dimensional vector of functions

Param var

a variable to differentiate with respect to

The function WronskianMatrix() calculates the Wronskian matrix of $$n$$ functions. The Wronskian matrix is created by putting each function as the first element of each column, and filling in the rest of each column by the $$(i-1)$$-th derivative, where $$i$$ is the current row. The Wronskian matrix is used to verify that the $$n$$ functions are linearly independent, usually solutions to a differential equation. If the determinant of the Wronskian matrix is zero, then the functions are dependent, otherwise they are independent.

Example

In> WronskianMatrix({Sin(x),Cos(x),x^4},x);
Out> {{Sin(x),Cos(x),x^4},{Cos(x),-Sin(x),4*x^3},
{-Sin(x),-Cos(x),12*x^2}};
In> PrettyForm(%)
/                                                 \
| ( Sin( x ) )      ( Cos( x ) )      /  4 \      |
|                                     \ x  /      |
|                                                 |
| ( Cos( x ) )      ( -( Sin( x ) ) ) /      3 \  |
|                                     \ 4 * x  /  |
|                                                 |
| ( -( Sin( x ) ) ) ( -( Cos( x ) ) ) /       2 \ |
|                                     \ 12 * x  / |
\                                                 /

The last element is a linear combination of the first two, so the determinant is zero:

In> A:=Determinant( WronskianMatrix( {x^4,x^3,2*x^4+3*x^3},x ) )
Out> x^4*3*x^2*(24*x^2+18*x)-x^4*(8*x^3+9*x^2)*6*x
+(2*x^4+3*x^3)*4*x^3*6*x-4*x^6*(24*x^2+18*x)+x^3
*(8*x^3+9*x^2)*12*x^2-(2*x^4+3*x^3)*3*x^2*12*x^2;
In> Simplify(A)
Out> 0;