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Higher Engineering Mathematics

John Bird

Chapter 33

Introduction to differential equations - all with Video Answers

Educators

Chapter Questions

02:39

Problem 1

Sketch the family of curves given by the equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=4 x$ and determine the equation of one of these curves which passes through the point $(2,3)$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:20

Problem 2

Determine the general solution of $x \frac{\mathrm{d} y}{\mathrm{~d} x}=2-4 x^{3}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:13

Problem 3

Find the particular solution of the differential equation $5 \frac{\mathrm{d} y}{\mathrm{~d} x}+2 x=3$, given the boundary conditions $y=1 \frac{2}{5}$ when $x=2$,

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:31

Problem 4

Solve the equation $2 t\left(t-\frac{\mathrm{d} \theta}{\mathrm{d} t}\right)=5$, given $\theta=2$ when $t=1$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:17

Problem 5

The bending moment $M$ of a beam is given by $\frac{\mathrm{d} M}{\mathrm{~d} x}=-w(l-x)$, where $w$ and $x$ are constants. Determine $M$ in terms of $x$ given: $M=\frac{1}{2} w l^{2}$ when $x=0$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:23

Problem 6

Find the general solution of
$$
\frac{\mathrm{d} y}{\mathrm{~d} x}=3+2 y
$$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:53

Problem 7

Determine the particular solution of $\left(y^{2}-1\right) \frac{\mathrm{d} y}{\mathrm{~d} x}=3 y$ given that $y=1$ when $x=2 \frac{1}{6}$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:43

Problem 8

(a) The variation of resistance, $R$ ohms, of an aluminium conductor with temperature $\theta^{\circ} \mathrm{C}$ is given by $\frac{\mathrm{d} R}{\mathrm{~d} \theta}=\alpha R$, where $\alpha$ is the temperature coefficient of resistance of aluminium. If $R=R_{0}$ when $\theta=0^{\circ} \mathrm{C}$, solve the equation for $R$. (b) If $\alpha=38 \times 10^{-4} /{ }^{\circ} \mathrm{C}$, determine the resistance of an aluminium conductor at $50^{\circ} \mathrm{C}$, correct to 3 significant figures, when its resistance at $0^{\circ} \mathrm{C}$ is $24.0 \Omega$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:45

Problem 9

Solve the equation $4 x y \frac{\mathrm{d} y}{\mathrm{~d} x}=y^{2}-1$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:07

Problem 10

Determine the particular solution of $\frac{\mathrm{d} \theta}{\mathrm{d} t}=2 \mathrm{e}^{3 t-2 \theta}$, given that $t=0$ when $\theta=0$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:54

Problem 11

Find the curve which satisfies the equation $x y=\left(1+x^{2}\right) \frac{\mathrm{d} y}{\mathrm{~d} x}$ and passes through the point $(0,1)$

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Varsha Aggarwal
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04:31

Problem 12

The current $i$ in an electric circuit containing resistance $R$ and inductance $L$ in series with a constant voltage source $E$ is given by the differential equation $E-L\left(\frac{\mathrm{d} i}{\mathrm{~d} t}\right)=R i$. Solve the equation and find $i$ in terms of time $t$ given that when $t=0, i=0$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:56

Problem 13

For an adiabatic expansion of a gas
$$
C_{v} \frac{\mathrm{d} p}{p}+C_{p} \frac{\mathrm{d} V}{V}=0
$$
where $C_{p}$ and $C_{v}$ are constants. Given $n=\frac{C_{p}}{C_{v}}$ show that $p V^{n}=$ constant

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator