Deck 11: First-Order Differential Equations

Question 1

State the onder and degree of the differential equation $y^{\prime \prime}+y^{5}=2 x^{3}$ .

A) order 2, degree 1
B) order 2, degree 3
C) order 5, degree 3
D) order 2, degree 5

Answer 1

order 2, degree 1

Question 2

Find the particular solution of the differential equation $x^{2} d y=y d x$ if $y=1$ when $x=1$ .

A) $\frac{1}{y}=-\frac{1}{x}+2$ or $y=\frac{x}{2 x-1}$
B) $\ln y=-\frac{1}{x}+1$ or $y=e^{\frac{-1}{x}+1}$
C) $\ln y=\ln x^{2}$ or $y=x^{2}$
D) $\ln y=-\frac{1}{x}+2$ or $y=e^{\frac{-1}{x}+2}$

Answer 2

$\ln y=-\frac{1}{x}+1$ or $y=e^{\frac{-1}{x}+1}$

Question 3

Find the general solution of the linear differential equation $y^{\prime}+3 y=e^{-3 x}$ .

A) $y=(x+C) e^{-3 x}$
B) $y=(x+C) e^{3 x}$
C) $y=1+C e^{3 x}$
D) $y=C e^{-3 x}$

Answer 3

$y=(x+C) e^{-3 x}$

Question 4

Find the particular solution of the differential equation $y^{\prime}-\frac{y}{x}=x^{2}$ if $y=4$ when $x=1$ .

A) $y=\frac{1}{2} x^{3}+\frac{7}{2} x$
B) $y=\frac{1}{2} x^{2}+\frac{7}{2}$
C) $y=x^{3}+7 x$
D) $y=\frac{1}{2} x+\frac{7}{2 x}$

Question 5

Find the equation for the current $i$ in a series circuit with inductance $L=0.1 \mathrm{H}$ , resistance $R=8$ ohms, and voltage $\mathrm{V}=12$ volts. The initial current $\mathrm{i} 0=2$ amperes when $\mathrm{t}=0$ .

A) $i=0.5 e^{-80 t}+1.5$ amperes
B) $\mathrm{i}=0.5 \mathrm{e}^{-80 \mathrm{t}}-1.5$ amperes
C) $i=2 e^{-80 t}+1.5$ amperes
D) $i=2 e^{-80 t}-1.5$ amperes

Question 6

A radioactive material with an original mass of $10 \mathrm{~g}$ has a mass of $8 \mathrm{~g}$ after 50 years. The material decays at a rate proportional to the amount present. Find an expression for the amount present at any time t. Also find its half-life.

A) $Q=10(0.8)^{\frac{t}{50}}$ or $Q=10 e^{-0.004463 t}$ ; its half- life is about 155 years.
B) $\mathrm{Q}=10(0.8)^{\frac{\mathrm{t}}{500}}$ or $\mathrm{Q}=10 \mathrm{e}^{-0.0004463 \mathrm{t}}$ ; its half- life is about 1553 years.
C) $\mathrm{Q}=10(8)^{\frac{\mathrm{t}}{500}}$ or $\mathrm{Q}=10 \mathrm{e}^{0.004159 t}$ ; its half- life is about 167 years.
D) $\mathrm{Q}=8(0.1)^{\frac{\mathrm{t}}{50}}$ or $\mathrm{Q}=8 \mathrm{e}^{-0.04605 \mathrm{t}}$ ; its half- life is about 15 years.

Question 7

State the order and degree of the differential equation ( $\left.y^{\prime \prime \prime}\right) 2-2 x y^{\prime}+5 y^{\prime \prime}=0$ .

Question 8

Verify that the function $y=f(x)$ is a solution of the differential equation $x \frac{d x}{d y}-y=x^{5}$ . $$f(x)=\frac{1}{4}\left(x^{5}+15 x\right)$$

Question 9

Solve the differential equation: $x(y-1) d x+\left(x^{2}+3\right) d y=0$

Question 10

Find the particular solution to the differential equation $\frac{d x}{d y}=\frac{y^{2}}{x^{3}}$ where $y=8$ when $x=4$ .

Question 11

For the problems below, solve each differential equation.
-11 $\frac{y d x}{x^{2}+y^{2}}+\frac{x d y}{x^{2}+y^{2}}+y^{2} d y$

Question 12

For the problems below, solve each differential equation.
-12 $\frac{x d y+y d x}{4 x^{3}}=d x$

Question 13

Find the particular solution of the differential equation $\frac{x d y-y d x}{x^{2}}=2 y d y$ where $\mathrm{y}=6$ when $\mathrm{x}=2$ .

Question 14

For the problems below, solve each differential equation.

- $\frac{d x}{d y}+6 x y=e^{-3 x^{2}} \sin x$

Question 15

For the problems below, solve each differential equation.

- $4 y d x+x d y=5 x^{2} d x$

Question 16

Radioactive material decays at a rate proportional to the amount present. For a certain radioactive substance, approximately $16 \%$ of the original quantity decomposes in 24 years. Find the half- life of this radioactive material.

Question 17

Find the equation for the current $\mathrm{i}$ in a series circuit with inductance $\mathrm{L}=0.2 \mathrm{H}$ , resistance $\mathrm{R}=120 \Omega$ , and voltage $\mathrm{V}=210$ volts if the initial current $\mathrm{i}_{0}=5$ amperes when $\mathrm{t}=0$ . (Hint: Use $\mathrm{L} \frac{\mathrm{di}}{\mathrm{dt}}+\mathrm{Ri}=\mathrm{V}$ .)

Question 18

An object at $450^{\circ} \mathrm{F}$ is cooled in air, which is at $75^{\circ} \mathrm{F}$ . If the object is $420^{\circ} \mathrm{F}$ after $6 \mathrm{~min}$ , find the temperature of the object after $45 \mathrm{~min}$ .