If the right side of the equation dy dx = f (x, y) can be expressed as a function of the ratio y/x only, then the equation is said to be homogeneous. Such equations can always be transformed into separable equations by a change of the dependent variable. The following method outline can be used for any homogeneous equation. That is, the substitution y = xv (x) transforms a homogeneous equation into a separable equation. The latter equation can be solved by direct integration, and then replacing v by y x gives the solution to the original equation. dy/dx = (x^2 + 5y^2) / 2xy

Question

Mathematics

Thomas Trujillo

18 August, 18:59

If the right side of the equation dy dx = f (x, y) can be expressed as a function of the ratio y/x only, then the equation is said to be homogeneous. Such equations can always be transformed into separable equations by a change of the dependent variable. The following method outline can be used for any homogeneous equation. That is, the substitution y = xv (x) transforms a homogeneous equation into a separable equation. The latter equation can be solved by direct integration, and then replacing v by y x gives the solution to the original equation. dy/dx = (x^2 + 5y^2) / 2xy

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Answers (1)

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Karli · Answer 1

y = x*sqrt (Cx - 1)

Step-by-step explanation:

Given:

dy / dx = (x^2 + 5y^2) / 2xy

Find:

Solve the given ODE by using appropriate substitution.

Solution:

- Rewrite the given ODE:

dy/dx = 0.5 (x/y) + 2.5 (y/x)

- use substitution y = x*v (x)

dy/dx = v + x*dv/dx

- Combine the two equations:

v + x*dv/dx = 0.5 * (1/v) + 2.5*v

x*dv/dx = 0.5 * (1/v) + 1.5*v

x*dv/dx = (v^2 + 1) / 2v

-Separate variables:

(2v. dv / (v^2 + 1) = dx / x

- Integrate both sides:

Ln (v^2 + 1) = Ln (x) + C

v^2 + 1 = Cx

v = sqrt (Cx - 1)

- Back substitution:

(y/x) = sqrt (Cx - 1)

y = x*sqrt (Cx - 1)