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

Accepted Answer

The order of a differential equation is the highest derivative present, which is 2 in this case ($y^{\prime \prime}$). The degree is the power of the highest order derivative when the equation is polynomial in derivatives, which is 1 here.

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 $y=e^{\frac{-1}{x}+2}$

Accepted Answer

B

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

Accepted Answer

The given differential equation is a first-order linear differential equation. The integrating factor is $e^{\int 3 \, dx} = e^{3x}$. Multiplying through by the integrating factor and solving gives the general solution $y = (x + C)e^{-3x}$.

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) $y=\frac{1}{2} x+\frac{7}{2 x}$

Accepted Answer

The differential equation is a first-order linear differential equation. Solving it using an integrating factor, we find the particular solution that satisfies the initial condition $y(1) = 4$ is $y=\frac{1}{2} x^{3}+\frac{7}{2} x$.

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

Accepted Answer

The equation for the current in an RL circuit is $i(t) = i_0 e^{-\frac{R}{L}t} + \frac{V}{R}(1 - e^{-\frac{R}{L}t})$. Substituting the given values, $i(t) = 2 e^{-80t} + \frac{12}{8}(1 - e^{-80t}) = 0.5 e^{-80t} + 1.5$.

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.

Accepted Answer

The rate of decay is proportional to the amount present, so we have the differential equation $$\frac{dQ}{dt} = kQ$$where $k$ is a constant. Solving this equation, we get $$Q(t) = Q_0 e^{kt}$$where $Q_0$ is the initial amount. Using the given information, we have $$8 = 10 e^{50k} \Rightarrow k = \frac{1}{50} \ln \frac{8}{10} = -0.004463$$Therefore, the amount present at any time $t$ is $$Q(t) = 10 e^{-0.004463 t}$$The half-life is the time it takes for the amount to decay to half of its original value. Therefore, we have $$5 = 10 e^{-0.004463 t_{1/2}} \Rightarrow t_{1/2} = \frac{1}{-0.004463} \ln \frac{5}{10} = 155 \text{ years}$$

Question 7

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

Accepted Answer

The answer of State the order and degree of the...

Question 8

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

Accepted Answer

The answer of Verify that the function $y=f(x)$ is a...

Question 9

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

Accepted Answer

The answer of Solve the differential equation: \(x(y-1) d x+\left(x^{2}+3\right)...

Question 10

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

Accepted Answer

The answer of Find the particular solution to the differential...

Question 11

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

Accepted Answer

The answer of For the problems below, solve each differential...

Question 12

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

Accepted Answer

The answer of For the problems below, solve each differential...

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 $\mathrm{x}=2$.

Accepted Answer

The answer of Find the particular solution of the differential...

Question 14

For the problems below, solve each differential equation.
- $\frac{d x}{d y}+6 x y=e^{-3 x^{2}} \sin x$

Accepted Answer

The answer of For the problems below, solve each differential...

Question 15

For the problems below, solve each differential equation.
-  $4 y d x+x d y=5 x^{2} d x$

Accepted Answer

The answer of For the problems below, solve each differential...

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.

Accepted Answer

The answer of Radioactive material decays at a rate proportional...

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 $\mathrm{L} \frac{\mathrm{di}}{\mathrm{dt}}+\mathrm{Ri}=\mathrm{V}$.)

Accepted Answer

The answer of Find the equation for the current $\mathrm{i}$...

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 $45 \mathrm{~min}$.

Accepted Answer

The answer of An object at $450^{\circ} \mathrm{F}$ is cooled...

Quiz 11: First-Order Differential Equations

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

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

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

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

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 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.

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

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

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

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

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

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

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 $\mathrm{x}=2$ .

For the problems below, solve each differential equation. - $\frac{d x}{d y}+6 x y=e^{-3 x^{2}} \sin x$

For the problems below, solve each differential equation. - $4 y d x+x d y=5 x^{2} d x$

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.

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 $45 \mathrm{~min}$ .

Quiz 11: First-Order Differential Equations

State the onder and degree of the differential equation y′′+y5=2x3y^{\prime \prime}+y^{5}=2 x^{3}y′′+y5=2x3 .

Find the particular solution of the differential equation x2dy=ydxx^{2} d y=y d xx2dy=ydx if y=1y=1y=1 when x=1x=1x=1 .

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

Find the particular solution of the differential equation y′−yx=x2y^{\prime}-\frac{y}{x}=x^{2}y′−xy​=x2 if y=4y=4y=4 when x=1x=1x=1 .

Find the equation for the current iii in a series circuit with inductance L=0.1HL=0.1 \mathrm{H}L=0.1H , resistance R=8R=8R=8 ohms, and voltage V=12\mathrm{V}=12V=12 volts. The initial current i0=2\mathrm{i} 0=2i0=2 amperes when t=0\mathrm{t}=0t=0 .

A radioactive material with an original mass of 10 g10 \mathrm{~g}10 g has a mass of 8 g8 \mathrm{~g}8 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.

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

Verify that the function y=f(x)y=f(x)y=f(x) is a solution of the differential equation xdxdy−y=x5x \frac{d x}{d y}-y=x^{5}xdydx​−y=x5 . f(x)=14(x5+15x)f(x)=\frac{1}{4}\left(x^{5}+15 x\right)f(x)=41​(x5+15x)

Solve the differential equation: x(y−1)dx+(x2+3)dy=0x(y-1) d x+\left(x^{2}+3\right) d y=0x(y−1)dx+(x2+3)dy=0

Find the particular solution to the differential equation dxdy=y2x3\frac{d x}{d y}=\frac{y^{2}}{x^{3}}dydx​=x3y2​ where y=8y=8y=8 when x=4x=4x=4 .

For the problems below, solve each differential equation. -11 ydxx2+y2+xdyx2+y2+y2dy\frac{y d x}{x^{2}+y^{2}}+\frac{x d y}{x^{2}+y^{2}}+y^{2} d yx2+y2ydx​+x2+y2xdy​+y2dy

For the problems below, solve each differential equation. -12 xdy+ydx4x3=dx\frac{x d y+y d x}{4 x^{3}}=d x4x3xdy+ydx​=dx

Find the particular solution of the differential equation xdy−ydxx2=2ydy\frac{x d y-y d x}{x^{2}}=2 y d yx2xdy−ydx​=2ydy where y=6\mathrm{y}=6y=6 when x=2\mathrm{x}=2x=2 .

For the problems below, solve each differential equation. - dxdy+6xy=e−3x2sin⁡x\frac{d x}{d y}+6 x y=e^{-3 x^{2}} \sin xdydx​+6xy=e−3x2sinx

For the problems below, solve each differential equation. - 4ydx+xdy=5x2dx4 y d x+x d y=5 x^{2} d x4ydx+xdy=5x2dx

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

An object at 450∘F450^{\circ} \mathrm{F}450∘F is cooled in air, which is at 75∘F75^{\circ} \mathrm{F}75∘F . If the object is 420∘F420^{\circ} \mathrm{F}420∘F after 6 min6 \mathrm{~min}6 min , find the temperature of the object after 45 min45 \mathrm{~min}45 min .

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

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

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

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

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 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.

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

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

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

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

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

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

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 $\mathrm{x}=2$ .

For the problems below, solve each differential equation. - $\frac{d x}{d y}+6 x y=e^{-3 x^{2}} \sin x$

For the problems below, solve each differential equation. - $4 y d x+x d y=5 x^{2} d x$

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.

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 $45 \mathrm{~min}$ .