Rivalry2

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Original 2026-01-01

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1 Euler’s theorem states that, if f(x, y, z) is a homogeneous function of degree n, then: x∂f∂x + y∂f∂y + z∂f∂z = n f(x, y, z) This means that if we multiply each variable by its partial derivative and then add them, we will get the degree n times the original function.

2 When a function is homogeneous, scaling its variable by a<a>number</a>t changes the whole function by tn. Euler’s theorem is a rule that comes from this scaling behavior. The steps given below explain Euler’s theorem:

3 Step 1: Assume f(x, y, z) is homogeneous of degree n. f(tx, ty, tz) = tn f(x, y, z)

4 Step 2: Differentiate both sides of the<a>equation</a>with respect to t, applying the derivative rule to each term.

5 Step 3: Apply the chain rule to the left-hand side: x∂f∂x + y∂f∂y + z∂f∂z

6 Step 4: Differentiate the right-hand side, ntn - 1f(x, y, z)

7 Step 5: Set t = 1, and you get, x∂f∂x + y∂f∂y + z∂f∂z = n f(x, y, z)

8 Hence, Euler's theorem is proved.

9 Homogeneous Differential Equation from Homogeneous Function

10 A homogeneous differential equation whose right-hand side is made from a homogeneous function of x and y. It can be written as:

11 dydx = f(x, y) If f(x, y) is a homogeneous function, then the equation is referred to as a homogeneous differential equation.

12 A function f(x, y) is called homogeneous of degree n if we can write it as:

13 f(x, y) = xn × gyx or f(x, y) = yn × gxy

14 The homogeneous function depends on the<a>ratio</a>of y to x or x to y.

15 A differential equation is called homogeneous when it can be written in the form of, dydx = g(x, y) This means the right side of the equation depends only on the ratio yx. The steps to solve it are provided below.

16 Step 1:We use a new variable: v = yx or y = v . x

17 Step 2:If y = v . x, we use the<a>product</a>rule to find dydx

18 dydx = v + xdvdx

19 Step 3:Substitute the value dydx into the given equation:

20 v + xdvdx = g(v)

21 Step 4:Rearrange to isolate dvdx

22 xdvdx = g(v) - v

23 Step 5:Separate the variables to rewrite the equation to separate v and x:

24 dvg(v) - v = dxx

25 Step 6: Integrate both sides,

26 1g(v) - vdv =1xdx + C

27 Here, C is the constant.

28 Step 7:Since v = yx, we substitute it back into the solution to get the answer in terms of x and y. If the equation is dxdy = f(x, y) is homogeneous, we use x = vy and solve it the same way.