Appendix A
Solving ordinary differential equations
Typical dynamical equations of Physics are
1) Force in the x-direction = mass × acceleration in the x-direction with the
mathematical form
Fx = max = md2x/dt2,
and
2) The amplitude y(x, t) of a wave at (x, t), travelling at constant speed V along
the x-axis with the mathematical form
(1/V2)∂2y/∂t2 – ∂2y/∂x2 = 0.
Such equations, that involve differential coefficients, are called differential equations.
An equation of the form
f(x, y(x), dy(x)/dx; ar) = 0 (A.1)
that contains
i) a variable y that depends on a single, independent variable x,
ii) a first derivative dy(x)/dx,
and
iii) constants, ar,
, 188 ORDINARY DIFFERENTIAL EQUATIONS
is called an ordinary (a single independent variable) differential equation of the first order
(a first derivative, only).
An equation of the form
f(x1, x2, ...x n, y(x1, x2, ...x n), ∂y/∂x1, ∂y/∂x2, ...∂y/∂xn; ∂2y∂x12, ∂2y/∂x22,
...∂2y/∂xn2; ∂ny/∂x1n, ∂ny/∂x2n, ...∂ny/∂xnn; a1, a2, ...a r) = 0 (A.2)
that contains
i) a variable y that depends on n-independent variables x1, x2, ...x n,
ii) the 1st-, 2nd-, ...nth-order partial derivatives:
∂y/∂x1, ...∂2y/∂x12, ...∂ny/∂x1n, ...,
and
iii) r constants, a1, a2, ...a r,
is called a partial differential equation of the nth-order.
Some of the techniques for solving ordinary linear differential equations are given in this
appendix.
An ordinary differential equation is formed from a particular functional relation,
f(x, y; a1, a2, ...a n) that involves n arbitrary constants. Successive differentiations of f with
respect to x, yield n relationships involving x, y, and the first n derivatives of y with respect
to x, and some (or possibly all) of the n constants. There are (n + 1) relationships from
which the n constants can be eliminated. The result will involve dny/dxn, differential
coefficients of lower orders, together with x, and y, and no arbitrary constants.
Consider, for example, the standard equation of a parabola:
Solving ordinary differential equations
Typical dynamical equations of Physics are
1) Force in the x-direction = mass × acceleration in the x-direction with the
mathematical form
Fx = max = md2x/dt2,
and
2) The amplitude y(x, t) of a wave at (x, t), travelling at constant speed V along
the x-axis with the mathematical form
(1/V2)∂2y/∂t2 – ∂2y/∂x2 = 0.
Such equations, that involve differential coefficients, are called differential equations.
An equation of the form
f(x, y(x), dy(x)/dx; ar) = 0 (A.1)
that contains
i) a variable y that depends on a single, independent variable x,
ii) a first derivative dy(x)/dx,
and
iii) constants, ar,
, 188 ORDINARY DIFFERENTIAL EQUATIONS
is called an ordinary (a single independent variable) differential equation of the first order
(a first derivative, only).
An equation of the form
f(x1, x2, ...x n, y(x1, x2, ...x n), ∂y/∂x1, ∂y/∂x2, ...∂y/∂xn; ∂2y∂x12, ∂2y/∂x22,
...∂2y/∂xn2; ∂ny/∂x1n, ∂ny/∂x2n, ...∂ny/∂xnn; a1, a2, ...a r) = 0 (A.2)
that contains
i) a variable y that depends on n-independent variables x1, x2, ...x n,
ii) the 1st-, 2nd-, ...nth-order partial derivatives:
∂y/∂x1, ...∂2y/∂x12, ...∂ny/∂x1n, ...,
and
iii) r constants, a1, a2, ...a r,
is called a partial differential equation of the nth-order.
Some of the techniques for solving ordinary linear differential equations are given in this
appendix.
An ordinary differential equation is formed from a particular functional relation,
f(x, y; a1, a2, ...a n) that involves n arbitrary constants. Successive differentiations of f with
respect to x, yield n relationships involving x, y, and the first n derivatives of y with respect
to x, and some (or possibly all) of the n constants. There are (n + 1) relationships from
which the n constants can be eliminated. The result will involve dny/dxn, differential
coefficients of lower orders, together with x, and y, and no arbitrary constants.
Consider, for example, the standard equation of a parabola:
