 # Homogeneous Form

## •    In order to understand homogeneous differential equations, it is important to understand homogeneous function. A function f(x, y) is called a homogeneous function of degree n, if a function f can be expressed as follows:•    Homogeneous Differential Equation: A differential equation of the form dy/dx = F(x, y) is said to be homogeneous differential equation if the function F(x, y) is a homogeneous function of degree zero.•    Solution of Homogeneous Differential Equation: o    Substitute y = vx and dy/dx = v + x (dv/dx) in the differential equation. o    Separate the variables and integrate on both sides. o    Replace v by y/x.•    Differential Equation Reducible to Homogeneous Form: •    A differential equation of the form dy/dx = [a1x + b1y + c1]/ [a2x + b2y + c2] can be reduced to homogeneous form by using the following transformations: x = X + h and y =Y + k, where, h and k are constants to be determined so that c1 and c2 are eliminated.•    There are various real life usages of differential equations. An ideal spring with a spring constant k is described by the simple harmonic oscillation. Newton's second law gives its equation of motion in the form of a homogeneous second-order linear differential equation.Keywords: Homogeneous differential equation, Homogeneous function

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• Q1 Marks:1

y = Acos x + Bsinx.

##### Explanation:

Clearly, y = Acos x + Bsinx  satisfies the given diferential equation.

• Q2

If y = y(x) and equals

Marks:1

1/3.

##### Explanation: • Q3

The solution of , where y(0) = 2 is

Marks:1 ##### Explanation: • Q4

The solution of the differential equation, , is

Marks:1 ##### Explanation: • Q5 Marks:1 We see that 