Question

4 Find the total differential of the function u = x^2y−3y
I. 2x+(x^2−3)dy
II. 2xydx+x^2dy
III. 2xydx+(x^2−3)dy
IV. 2xydx+(x^3−2)dy
5 The total differential du of a function u (x, y) = 0 is defined as
I. ∂u/∂x dx+∂u.∂x dy=0
II. ∂u/∂x dy+∂u/∂x dy=0
III. ∂u/∂x dx+∂u/∂y dx=0
IV. ∂u/∂x dx+∂u/∂y dy=0
6 A differential equation involving only a single independent variable is called ………… equation.
I. extraordinary differential
II. ordinary differential
III. super-ordinary differential
IV. partial differential
7 The homogeneous equation d/ydx=x^4+x^3y+y^4/3x^3y+y^4 is of …… degree
I. first , II. second , III. third , IV. fourth

Answer 1

ANSWER on Question #59453 – Math – Differential Equations

QUESTION 4

Find the total differential of the function $u = x^{2}y - 3y$

a) $2xdx + (x^{2} - 3)dy$

b) $2xydx + x^{2}dy$

c) $2xydx + (x^{2} - 3)dy$

d) $2xydx + (x^{3} - 2)dy$

SOLUTION

By the definition, for any function $u(x,y)$

$$du = \frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy$$

In our case

$$u = x^{2}y - 3y \iff \left\{ \begin{array}{l} \frac{\partial u}{\partial x} = \frac{\partial}{\partial x} (x^{2}y - 3y) = 2xy \\ \frac{\partial u}{\partial y} = \frac{\partial}{\partial y} (x^{2}y - 3y) = x^{2} - 3 \end{array} \right.$$

That is why

$$du = \frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy = 2xy dx + (x^{2} - 3) dy$$

**ANSWER**: c) $2xy dx + (x^{2} - 3) dy$.

QUESTION 5

The total differential du of a function $u(x,y) = 0$ is defined as

a) $\frac{\partial u}{\partial x} dx + \partial u \cdot \partial x \, dy = 0$

b) $\frac{\partial u}{\partial x} dy + \frac{\partial u}{\partial y} dy = 0$

c) $\frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dx = 0$

d) $\frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy = 0$

SOLUTION

$$u(x, y) = 0 \Leftrightarrow d(u(x, y)) = 0$$ $$du = \frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy$$ $$du = 0 \Leftrightarrow \frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy = 0$$

**ANSWER**: d) $\frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy = 0$.

QUESTION 6

A differential equation involving only a single independent variable is called ... equation.

a) extraordinary differential

b) ordinary differential

c) super-ordinary differential

d) partial differential

SOLUTION

By definition, the differential equation with one independent variable is called an ordinary differential equation

**ANSWER**: b) ordinary differential.

QUESTION 7

The homogeneous equation $\frac{dy}{dx} = x^4 + x^3 y + \frac{y^4}{3x^3 y} + y^4$ is of ... degree

a) first

b) second

c) third

d) fourth

SOLUTION

It is an ordinary differential equation, because we see only one independent variable $x$ (the derivative is taken with respect to it) and one dependent function $y(x)$. By definition, the degree of differential equation is determined by the highest derivative.

Therefore, this equation is of first degree.

**ANSWER : a) first.**

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