Solved on Feb 05, 2024

Find the equation of a curve with $\frac{dy}{dx} = 42yx^{13}$ and $y$ -intercept at 2.

STEP 1

Assumptions
1. The given differential equation is $\frac{d y}{d x}=42 y x^{13}$ .
2. The curve has a $y$ -intercept of 2, which means $y(0) = 2$ .

STEP 2

To find the equation of the curve, we need to solve the differential equation. This is a first-order linear differential equation which can be solved by separation of variables.

STEP 3

Separate the variables $y$ and $x$ to opposite sides of the equation.
$\frac{1}{y} \frac{d y}{d x} = 42 x^{13}$

STEP 4

Integrate both sides of the equation with respect to $x$ .
$\int \frac{1}{y} \frac{d y}{d x} dx = \int 42 x^{13} dx$

STEP 5

The left side of the equation simplifies to the natural logarithm of the absolute value of $y$ , and the right side integrates to $\frac{42}{14} x^{14}$ plus a constant of integration $C$ .
$\ln |y| = 3 x^{14} + C$

STEP 6

Exponentiate both sides to solve for $y$ .
$|y| = e^{3 x^{14} + C}$

STEP 7

Since we are looking for a particular solution where $y$ is positive (because the $y$ -intercept is positive), we can drop the absolute value.
$y = e^{3 x^{14}} \cdot e^C$

STEP 8

Recognize that $e^C$ is just another constant, which we can call $A$ .
$y = A e^{3 x^{14}}$

STEP 9

Use the initial condition $y(0) = 2$ to find the value of $A$ .
$2 = A e^{3 \cdot 0^{14}}$

STEP 10

Since $e^0 = 1$ , we can solve for $A$ .
$A = 2$

STEP 11

Substitute the value of $A$ back into the equation for $y$ .
$y = 2 e^{3 x^{14}}$
This is the equation of the curve that satisfies the given differential equation and whose $y$ -intercept is 2.
$y(x) = 2 e^{3 x^{14}}$

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Find the equation of a curve with $\frac{dy}{dx} = 42yx^{13}$ and $y$ -intercept at 2.

STEP 1

Assumptions
1. The given differential equation is $\frac{d y}{d x}=42 y x^{13}$ .
2. The curve has a $y$ -intercept of 2, which means $y(0) = 2$ .

STEP 2

To find the equation of the curve, we need to solve the differential equation. This is a first-order linear differential equation which can be solved by separation of variables.

STEP 3

Separate the variables $y$ and $x$ to opposite sides of the equation.
$\frac{1}{y} \frac{d y}{d x} = 42 x^{13}$

STEP 4

Integrate both sides of the equation with respect to $x$ .
$\int \frac{1}{y} \frac{d y}{d x} dx = \int 42 x^{13} dx$

STEP 5

The left side of the equation simplifies to the natural logarithm of the absolute value of $y$ , and the right side integrates to $\frac{42}{14} x^{14}$ plus a constant of integration $C$ .
$\ln |y| = 3 x^{14} + C$

STEP 6

Exponentiate both sides to solve for $y$ .
$|y| = e^{3 x^{14} + C}$

STEP 7

Since we are looking for a particular solution where $y$ is positive (because the $y$ -intercept is positive), we can drop the absolute value.
$y = e^{3 x^{14}} \cdot e^C$

STEP 8

Recognize that $e^C$ is just another constant, which we can call $A$ .
$y = A e^{3 x^{14}}$

STEP 9

Use the initial condition $y(0) = 2$ to find the value of $A$ .
$2 = A e^{3 \cdot 0^{14}}$

STEP 10

Since $e^0 = 1$ , we can solve for $A$ .
$A = 2$

STEP 11

Substitute the value of $A$ back into the equation for $y$ .
$y = 2 e^{3 x^{14}}$
This is the equation of the curve that satisfies the given differential equation and whose $y$ -intercept is 2.
$y(x) = 2 e^{3 x^{14}}$

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Find the equation of a curve with dydx=42yx13\frac{dy}{dx} = 42yx^{13}dxdy​=42yx13 and yyy-intercept at 2.

STEP 1

Assumptions1. The given differential equation is dydx=42yx13\frac{d y}{d x}=42 y x^{13}dxdy​=42yx13.2. The curve has a yyy-intercept of 2, which means y(0)=2y(0) = 2y(0)=2.

STEP 2

To find the equation of the curve, we need to solve the differential equation. This is a first-order linear differential equation which can be solved by separation of variables.

STEP 3

Separate the variables yyy and xxx to opposite sides of the equation.1ydydx=42x13\frac{1}{y} \frac{d y}{d x} = 42 x^{13}y1​dxdy​=42x13

STEP 4

Integrate both sides of the equation with respect to xxx.∫1ydydxdx=∫42x13dx\int \frac{1}{y} \frac{d y}{d x} dx = \int 42 x^{13} dx∫y1​dxdy​dx=∫42x13dx

STEP 5

The left side of the equation simplifies to the natural logarithm of the absolute value of yyy, and the right side integrates to 4214x14\frac{42}{14} x^{14}1442​x14 plus a constant of integration CCC.ln⁡∣y∣=3x14+C\ln |y| = 3 x^{14} + Cln∣y∣=3x14+C

STEP 6

Exponentiate both sides to solve for yyy.∣y∣=e3x14+C|y| = e^{3 x^{14} + C}∣y∣=e3x14+C

STEP 7

Since we are looking for a particular solution where yyy is positive (because the yyy-intercept is positive), we can drop the absolute value.y=e3x14⋅eCy = e^{3 x^{14}} \cdot e^Cy=e3x14⋅eC

STEP 8

Recognize that eCe^CeC is just another constant, which we can call AAA.y=Ae3x14y = A e^{3 x^{14}}y=Ae3x14

STEP 9

Use the initial condition y(0)=2y(0) = 2y(0)=2 to find the value of AAA.2=Ae3⋅0142 = A e^{3 \cdot 0^{14}}2=Ae3⋅014

STEP 10

Since e0=1e^0 = 1e0=1, we can solve for AAA.A=2A = 2A=2

STEP 11

Substitute the value of AAA back into the equation for yyy.y=2e3x14y = 2 e^{3 x^{14}}y=2e3x14This is the equation of the curve that satisfies the given differential equation and whose yyy-intercept is 2.y(x)=2e3x14y(x) = 2 e^{3 x^{14}}y(x)=2e3x14

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the equation of a curve with $\frac{dy}{dx} = 42yx^{13}$ and $y$ -intercept at 2.

Assumptions
1. The given differential equation is $\frac{d y}{d x}=42 y x^{13}$ .
2. The curve has a $y$ -intercept of 2, which means $y(0) = 2$ .

Separate the variables $y$ and $x$ to opposite sides of the equation.
$\frac{1}{y} \frac{d y}{d x} = 42 x^{13}$

Integrate both sides of the equation with respect to $x$ .
$\int \frac{1}{y} \frac{d y}{d x} dx = \int 42 x^{13} dx$

The left side of the equation simplifies to the natural logarithm of the absolute value of $y$ , and the right side integrates to $\frac{42}{14} x^{14}$ plus a constant of integration $C$ .
$\ln |y| = 3 x^{14} + C$

Exponentiate both sides to solve for $y$ .
$|y| = e^{3 x^{14} + C}$

Since we are looking for a particular solution where $y$ is positive (because the $y$ -intercept is positive), we can drop the absolute value.
$y = e^{3 x^{14}} \cdot e^C$

Recognize that $e^C$ is just another constant, which we can call $A$ .
$y = A e^{3 x^{14}}$

Use the initial condition $y(0) = 2$ to find the value of $A$ .
$2 = A e^{3 \cdot 0^{14}}$

Since $e^0 = 1$ , we can solve for $A$ .
$A = 2$

Substitute the value of $A$ back into the equation for $y$ .
$y = 2 e^{3 x^{14}}$
This is the equation of the curve that satisfies the given differential equation and whose $y$ -intercept is 2.
$y(x) = 2 e^{3 x^{14}}$