Question

8 One of these is a homogeneous equation

I. h(x,y)=x^3+2xy+3xy^2+4y^3

II. h(x,y)=x^2+2x^2y+3xy^2+4y^3

III. h(x,y)=x^3+2x^2y+3xy^2+4y^3

IV. h(x,y)=x^3+2xy+3xy^2+4y^2
9 One of the following is not a separable equation

i. dy/dx=e^x+y

II. dy/dx=e^xy

III. dy/dx=x^2(y+y^2)

IV. (1+y^2)dx+(1+x^3)dy

Answer 1

Answer on Question #59454 – Math – Differential Equations

Question

8. One of these is a homogeneous equation

I. $h(x, y) = x^3 + 2xy + 3xy^2 + 4y^3$

II. $h(x, y) = x^2 + 2x^2y + 3xy^2 + 4y^3$

III. $h(x, y) = x^3 + 2x^2y + 3xy^2 + 4y^3$

IV. $h(x, y) = x^3 + 2xy + 3xy^2 + 4y^2$

Solution

The equation $M(x, y)dx + N(x, y)dy = 0$ is a homogeneous type if $M(x, y)$ and $N(x, y)$ are homogeneous functions of the same degree $n$ and homogeneous function of the degree $n$ is a function which satisfies the condition $f(tx, ty) = t^n f(x, y)$.

Check that the function III. $h(x, y) = x^3 + 2x^2y + 3xy^2 + 4y^3$ is homogeneous of order 3:

$h(tx, ty) = (tx)^3 + 2(tx)^2ty + 3tx(ty)^2 + 4(ty)^3 = t^3x^3 + 2t^3x^2y + 3y^3xy^2 + 4t^3y^3 = t^3(x^3 + 2x^2y + 3xy^2 + 4y^3) = t^3h(x, y)$.

Answer: III. $h(x, y) = x^3 + 2x^2y + 3xy^2 + 4y^3$.

Question

9. One of the following is not a separable equation

I. $dy/dx = e^x + y$

II. $dy/dx = e^xy$

III. $dy/dx = x^2(y + y^2)$

IV. $(1 + y^2)dx + (1 + x^3)dy$

Solution

The differential equation of the form $\frac{dy}{dx} = f(x, y)$ is called separable, if $f(x, y) = h(x) g(y)$.

Consider I. $dy/dx = e^x + y$, $f(x, y) = e^x + y = h(x) + g(y)$ is the sum of functions, so this equation is not a separable equation.

Answer: I. $dy/dx = e^x + y$.

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