• 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
y = Acos x + Bsinx.
Clearly, y = Acos x + Bsinx satisfies the given diferential equation.
If y = y(x) and equalsMarks:1
The solution of , where y(0) = 2 isMarks:1
The solution of the differential equation, , isMarks:1
By Hit and Trial Method(Checking each option by substituting in given equation),
We see that
Satisfies the given differential equation.