Check whether the differential equation $(x y) d x - (x^{3} + y^{3}) d y = 0$ is homogenous or not ?

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Solution

$(x y) d x - (x^{2} + y^{2}) d y = 0 \Rightarrow \frac{d y}{d x} = \frac{x y}{x^{2} + y^{2}} = f (x, y)$

Now $f (k x, k y) = \frac{k^{2} y}{k^{3} (x^{2} + y^{2})} = k^{- 1} f (x, y)$

So, it is a homogenous differential equation

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Similar questions

Which of the following is a homogeneous differential equation?
(a) (4x +6y+5)dy-(3y+2x+4)dx=0
(b) $x y d x - (x^{3} + y^{3}) d y = 0$
(c) $(x^{3} + 2 y^{2}) d x + 2 x y d y = 0$
(d) $y^{2} d x + (x^{2} - x y - y^{2}) d y = 0$

Q. Solve the differential equation :

(x^{3} + y^{3}) d y - x^{2} y d x = 0

Q. The solution of differential equation

x^{2} y d x - (x^{3} + y^{3}) d y = 0

Q. Does the differential equation

y^{3} d y + (x + y^{2}) d x = 0

becomes homogeneous if we put

y^{2} = t

Q. If

y

is the soultion of the differential equation

y^{3} \frac{d y}{d x} + x^{3} = 0, y (0) = 1,

the value of

y (- 1)

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Check whether the differential equation (xy)dx−(x3+y3)dy=0 is homogenous or not ?

(xy)dx−(x2+y2)dy=0⇒dydx=xyx2+y2=f(x,y) Nowf(kx,ky)=k2yk3(x2+y2)=k−1f(x,y) So, it is a homogenous differential equation

Check whether the differential equation $(x y) d x - (x^{3} + y^{3}) d y = 0$ is homogenous or not ?

$(x y) d x - (x^{2} + y^{2}) d y = 0 \Rightarrow \frac{d y}{d x} = \frac{x y}{x^{2} + y^{2}} = f (x, y)$

Now $f (k x, k y) = \frac{k^{2} y}{k^{3} (x^{2} + y^{2})} = k^{- 1} f (x, y)$

So, it is a homogenous differential equation