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Natural Sciences Notes Mathematics for Natural Sciences Notes

Ordinary Differential Equations Notes

Updated Ordinary Differential Equations Notes

Mathematics for Natural Sciences Notes

Mathematics for Natural Sciences

Approximately 176 pages

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Notes for Ordinary Differential Equations

General properties of ODEs:

Ordinary differential equations (ODEs) relate the value of some variable y to the values of its ordinary derivatives with respect to one other variable x.

Order of differential equations: The order of the highest derivative of y with respect to x:

First-order ODE: Second-order ODE: nth-order ODE:

Form of equation:

F(y,y,x)=0

Form of solution:

f(y,x)=const.

Form of equation:

F(y,y,x)=0

Form of solution:

f(y,x)=const.

Form of equation:

F(y(n),yn1,…,y,x)=0

Form of solution:

f(y,x)=const.

Solutions and conditions:

In the solutions of ODEs, an arbitrary constant of integration appears. Each value of this constant gives a different unique solution to the equation:

  • The expression for the solution including the arbitrary constant is called the general solution

  • If a condition is given to specify a particular value of the constant, the expression for this solution is called a unique solution

Two types of condition are generally encountered:

  • An initial condition is the value of the function at t=0, when one of the variables is time.

  • A boundary condition is the value of the function at x=0, when one of the variables is position.

First-order ordinary differential equations:

Each different type of first-order ODE requires its own method of solution, as follows.

Directly-integrable, first-order: Method of solution:
$$\boxed{\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{= f(x)}}$$

Integrate directly to produce y=f(x)dx such that the problem is reduced to finding the integral of a single-variable function. By the Fundamental Theorem of Calculus, the solution is:

$$\boxed{y = F(x) + c}$$

Which is a family of solutions for which each member is distinguished by a unique value of the parameter c.

Separable, first-order: Method of solution:
$$\boxed{\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{\mathbf{f}\left( \mathbf{x} \right)}{\mathbf{g}\left( \mathbf{y} \right)}}$$

Multiply by g(y) to produce a directly-integrable ODE:

$$\int_{}^{}{g(y)\frac{dy}{dx}}dx = \int_{}^{}{f(x)}dx\therefore\int_{}^{}{g(y)}dy = \int_{}^{}{f(x)}dx$$

$$\boxed{G(y) = F(x) + c}$$

Which is a family of solutions for which each member is distinguished by a unique value of the parameter c.

Linear, first-order:

  • A function of a single variable y is linear if 1 is the highest power to which y is raised, for example f(y)=ax+b or f(y)=dy/dx+cy+d

  • An operator L is linear if it satisfies the condition that: L(ax+by)=aL(x)+bL(y), for constants a and b

  • The operator L=d/dx is linear when acting on differentiable functions of x, while the operator dx is linear when acting on integrable functions of x

  • The operator for linear first-order ODEs is:

$$L = \frac{d}{dx} + p(x)$$

For some function p(x) of x.

  • Linear first-order ODEs can be divided into two categories: homogeneous for Ly=0 and inhomogeneousfor Ly=f(x).

Homogeneous, linear, first-order: Method of solution:
$$\boxed{\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+ p}\left( \mathbf{x} \right)\mathbf{y = 0}}$$

This is a separable first-order ODE:

$$\frac{dy}{dx} = - p(x)y\therefore\int_{}^{}\frac{dy}{y} = - \int_{}^{}{p(x)}dx\therefore\ln y = - \int_{}^{}{p(x)}dx$$

y=ep(x)dx

$$\boxed{y = e^{- P(x) + c}}$$

Which is a family of solutions for which each member is distinguished by a unique value of the parameter c.

Inhomogeneous, linear, first-order: Method of solution:
$$\boxed{\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+ p}\left( \mathbf{x} \right)\mathbf{y =}\mathbf{f(x)}}$$

Reduce to a separable first-order ODE by multiplying by an integrating factor μ(x):

$$\mu(x)\frac{dy}{dx} + \mu(x)p(x)y = \mu(x)f(x)$$

μ(x) is chosen so that the left-hand side of this equation is a derivative with respect to x, such that the equation becomes separable μ(x) therefore satisfies the condition that:

$$\mu(x)\frac{dy}{dx} = \frac{d}{dx}\left( \mu(x)y \right) - y\frac{d}{dx}\left( \mu(x) \right)$$

$$\therefore LHS = \frac{d}{dx}\left( \mu(x)y \right) - y\frac{d}{dx}\left( \mu(x) \right) + \mu(x)p(x)y\therefore y\frac{d}{dx}\left( \mu(x) \right) = \mu(x)p(x)y$$

$$\therefore\frac{d}{dx}\left( \mu(x) \right) = \mu(x)p(x)\therefore\frac{d\mu}{dx} = \mu(x)p(x)$$

$$\therefore\int_{}^{}\frac{d\mu}{\mu} = \int_{}^{}{p(x)}dx$$

μ(x)=exp(x)dx

Method of solution:

This integrating factor can then be substituted back into the original ODE to give:

$$\mu(x)\frac{dy}{dx} + \mu(x)p(x)y = \mu(x)f(x)\therefore\frac{d}{dx}\left( y\mu(x) \right) = \mu(x)f(x)$$

yμ(x)=μ(x)f(x)dx

$$\boxed{y = \frac{1}{\mu(x)}\int_{}^{}{\mu(x)}f(x)dx}$$

Which is a family of solutions for which each member is distinguished by a unique value of the parameter c (not shown).

Solution spaces for linear, first-order ODEs:

Homogeneous solution space: Inhomogeneous solution space:

Linear, thus solutions satisfy the principle of superposition:

For solutions y1 and y2:

$$\frac{dy_{1}}{dx} + p(x)y_{1} = 0,\ \ \frac{dy_{2}}{dx} + p(x)y_{2} = 0$$

A linear combination of these solutions is y=ay1+by2:

$$\frac{d\left( ay_{1} + by_{2} \right)}{dx} + p(x)\left( ay_{1} + by_{2} \right)$$

$$= a\left( \frac{dy_{1}}{dx} + p(x)y_{1} \right) + b\left( \frac{dy_{2}}{dx} + p(x)y_{2} \right)$$

=0+0=0

Thus ay1+by2 is also a solution; the solutions are linear and superimposable.

Non-linear, thus solutions do not satisfy the principle of superposition:

For solutions y1 and y2:

$$\frac{dy_{1}}{dx} + p(x)y_{1} = f(x),\ \ \frac{dy_{2}}{dx} + p(x)y_{2} = f(x)$$

A linear combination of these solutions is y=ay1+by2:

$$\frac{d\left( ay_{1} + by_{2} \right)}{dx} + p(x)\left( ay_{1} + by_{2} \right)$$

$$= a\left( \frac{dy_{1}}{dx} + p(x)y_{1} \right) + b\left( \frac{dy_{2}}{dx} + p(x)y_{2} \right)$$

=af(x)+bf(x)f(x)

Thus ay1+by2 is not a solution; the solutions are not linear and superimposable.

Substitutions for first-order ODEs:

Many non-linear or non-separable first-order ODEs can be reduced to linear or separable ODEs...

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