Quiz about Week 9 0B2 OLD

Question 1

Let $$m\ $$ be a natural number. What does it mean for a function N(x,y) to be a homogeneous function of degree m?

Accepted Answer

a) $$N\left(\lambda x,\lambda y\right)=\lambda^mN\left(x,y\right)$$ for any number $$\lambda$$ .

Answer

b) $$N\left(\lambda x,\lambda y\right)=\lambda N\left(x,y\right)$$ for any number $$\lambda$$ .

Answer

c) $$N\left(\lambda x,\lambda y\right)=\lambda^m$$ for any number $$\lambda$$ .

Answer

d) All the terms of N(x,y) have total degree m.

Question 2

What is a homogeneous ODE?

Accepted Answer

b) An ODE that can be written in the form $$\frac{ ext{d}y}{ ext{d}x}=\frac{N\left(x,y ight)}{M\left(x,y ight)}$$ where M and N are both homogeneous functions of the same degree.

Answer

a) An ODE that can be written in the form $$\frac{ ext{d}y}{ ext{d}x}=\frac{N\left(x,y ight)}{M\left(x,y ight)}$$ where M and N are both homogeneous functions of different degrees.

Answer

c) An ODE that can be written in the form $$\frac{ ext{d}y}{ ext{d}x}=\frac{N\left(x,y ight)}{M\left(x,y ight)}$$ where M and N are both homogeneous functions.

Answer

d) An ODE that can be written in the form $$\frac{ ext{d}y}{ ext{d}x}=\frac{N\left(x,y ight)}{M\left(x,y ight)}$$ where M and N are both homogeneous functions such that $$N\left(\lambda x,\lambda y ight)=M\left(\lambda x,\lambda y ight)$$ for all values of $$\lambda$$ .

Question 3

Why does the substitution y=vx, where v is another variable that depends on x, help to solve homogeneous ODEs.

Accepted Answer

d) It converts the ODE into a separable one in the variables x and v.

Answer

a) It is a solution of the ODE.

Answer

b) It converts the ODE into a directly integrable one in the variables x and v.

Answer

c) It converts the ODE into a first order one in the variables x and v.

Question 4

Suppose that we have an ODE $$\frac{dy}{dx}=F\left(\frac{y}{x}\right)$$ where F(v) is a function. Is it true that this ODE is necessarily homogeneous?

Accepted Answer

Answer

Question 5

Suppose that we have an ODE $$\frac{dy}{dx}=G\left(x,y ight)$$ where the G(x,y) function which is not is a homogeneous function of degree 0 i.e. there are some real numbers a,b and $$\lambda$$ such that $$G(\lambda a,\lambda b) e G(a,b)$$ . What can we then say about the ODE?

Accepted Answer

c) It is not homogeneous.

Answer

a) It is homogeneous.

Answer

b) It is separable.

Answer

d) It is not separable.

Question 6

hich of the options below express the general solution of the ODE

$$\frac{dy}{dx}=\frac{x}{2y^2}$$ ?

In all of the options below, C denotes a constant.

Accepted Answer

c) $$y(x)=\sqrt[3]{\frac{3}{4}x^2+C}$$

Accepted Answer

d) $$y(x)=\sqrt[3]{\frac{3}{4}\left(x^2+C\right)}$$

Answer

a) $$y(x)=\left(\sqrt[3]{\frac{3}{2}\left(x^2+C\right)}\right)^{-1}$$

Answer

b) $$y(x)=\sqrt[3]{\frac{1}{3}\left(x^2+C\right)}$$

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