Question Type 3: Homogeneous differential equations (substitution 𝑣 = 𝑦 / 𝑥 ) Exercises

By substituting $y = vx$, solve the differential equation

$\frac{dy}{dx} = \frac{5x - 2y}{2x + 5y}.$

Solve the differential equation

$\frac{dy}{dx} = \frac{x - y}{x + y}.$

By using the substitution $y=vx$, solve the differential equation

$\frac{dy}{dx} = \biggl(\frac{y}{x}\biggr)^{1/2}.$

Solve the differential equation

$\frac{dy}{dx} = \frac{3y - x}{y + x}$

Solve the differential equation

$\frac{dy}{dx} = \frac{2x - y}{x + 2y}$

Solve the differential equation

$\frac{dy}{dx} = \frac{y}{x} + 1.$

Solve the initial-value problem: $\frac{dy}{dx} = \frac{y}{x} + 2, \quad y(1)=3$

Find the general solution of the differential equation

$\frac{dy}{dx} = \frac{x + y}{x - y}.$

[6 marks]

Solve the differential equation

Solve the differential equation

$\frac{dy}{dx} = \frac{y - 2x}{y + 2x}.$

Solve the differential equation

$\frac{dy}{dx} = \frac{y^2 + x^2}{xy}.$

Solve the initial-value problem

$\frac{dy}{dx} = \frac{2y - x}{y - 2x}, \quad y(1)=1$