Homogeneous Differential Equations

Homogeneous differential equations

Learning objective: to recognise and solve homogeneous differential equations

Before we begin with today's lesson

What do you remember from last time?

Multiple Choice

Which of the following best describes what a differential equation is?

• A differential equation describes the relationship between a function and its derivatives.
• A differential equation is a type of algebraic equation.
• A differential equation only involves constants and not variables.
• A differential equation describes the relationship between two unrelated functions.

Multiple Select

Which of the following are examples of differential equations?

• x^2 + y = 5
• sin(x) + cos(y) = 0
• y = mx + b
• dy/dx = 3x^2

Multiple Choice

What is the mathematical definition of a separable differential equation?

• A separable differential equation is always linear in y.
• A separable differential equation cannot be expressed in any form.
• A separable differential equation can be expressed as dy/dx = g(x)h(y), allowing separation of variables.
• A separable differential equation involves only constant coefficients.

Multiple Choice

Solve the following differential equation: $$\left(x^2+5\right)\times\ \frac{dy}{dx}=\frac{2x}{y^2}$$

• y=(5x+C)^(1/3)

• y = (2ln|x^2 + 5| + C)^(1/2)

• y = (x^3 + C)^(1/2)
• y = (3ln|x^2 + 5| + C)^(1/3)

Definition of homogeneous DE

​Example

Multiple Choice

Which of the following is a homogeneous differential equations?

• dy/dx = (x+y)/(x-y)

• dy/dx = 3y + 5x
• dy/dx = x^2 + y^2
• dy/dx = y + 2

Multiple Choice

Which of the following is a homogeneous differential equation?

• dy/dx = y/(3x+1)

• 4+dy/dx = 2y

• dy/dx = (y+2)/(x-1)

• dy/dx = (x-y)/x

Multiple Choice

Which of the following is a homogeneous differential equation?

• $$\frac{dy}{dx}=xy^2-2y^2$$

• $$\frac{dy}{dx}=\frac{x}{y^2}$$

• $$\frac{dy}{dx}=\frac{y^2-x^2}{2xy}$$

• $$\frac{dy}{dx}=y\sin\left(x\right)$$

​How to solve homogeneous DE?

Example

Multiple Choice

Solve $$\frac{dy}{dx}=\frac{x-y}{x}$$ given that when $$y=2$$ , $$x=1$$

• y = x + 1/x
• y = x^2 + 1
• y = (x^2+3) / (2x)

• y = 2x

Multiple Choice

Solve dy/dx = (x + 2y) / (4x) given that y(1) = 2

• y = (3/4)x + 2
• y = (1/4)x + 1
• y = (1/2)x + 3/2 * sqrt(x)

• y = (1/2)x + 1

Multiple Choice

Solve dy/dx = (xy + y^2) / x^2

• y = x/(C - ln|x|)

• y = Cx^2
• y = x^2 + C
• y = C/x

Multiple Choice

Solve dy/dx = cos^2 ( y / x ) + y / x

• y = x tan( ln |x| + C )

• y = x arctan( ln|x| + C )

• y = x arctan( 1/x + C )

• y = x arcsin ( 1/x + C)