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Differential Equations — Part 2: Homogeneous and linear equations

Beta

Lesson

Differential Equations, Part 2

Two new types, two new tools.

In this lesson you will

Test a function for homogeneity and find its degree
Solve homogeneous equations with the substitution y = vx
Solve linear equations dy/dx + Py = Q with an integrating factor
Solve equations linear in x and choose the right method

$C = 0$
$C = 1$
$C = 2$

Solutions of $\cos x\dfrac{dy}{dx} + y\sin x = \cos^2 x$: one family, one curve through each starting point.

Concept

What you will be able to do

Test whether a function is homogeneous
Solve homogeneous equations with $y = vx$
Solve linear equations with an integrating factor
Handle equations linear in $x$ , and choose a method

Part 2 on one card.
Type	Recognise it by	Method
Homogeneous	$F(\lambda x, \lambda y) = F(x, y)$	$y = vx$ , then separate
Linear in $y$	$\dfrac{dy}{dx} + Py = Q$ , $P, Q$ in $x$	I.F. $= e^{\int P\,dx}$
Linear in $x$	$\dfrac{dx}{dy} + Px = Q$ , $P, Q$ in $y$	I.F. $= e^{\int P\,dy}$

Concept

Homogeneous functions

$F(x, y)$ is homogeneous of degree $n$ if $F(\lambda x, \lambda y) = \lambda^n F(x, y)$ for all non-zero $\lambda$ .

Replace $x, y$ by $\lambda x, \lambda y$ and try to pull out $\lambda^n$.
$F(x, y)$	$F(\lambda x, \lambda y)$	Homogeneous?
$\dfrac{x^2 + 3y^2}{xy}$	$\lambda^0F(x, y)$	yes, degree $0$
$\sqrt{x^2 + y^2}$	$\lambda F(x, y)$	yes, degree $1$
$\dfrac{x^2y + y^3}{x + y}$	$\lambda^2F(x, y)$	yes, degree $2$
$x^2 + y$	$\lambda^2x^2 + \lambda y$	no