integrate the following
(3y-7x-3)dx+(7y-3x-7)dy=0
rearrange things a it to get
(3y-7x-3)+(7y-3x-7)y'=0
now integrate term by term
3xy-7/2x^2-3x + 7/2y^2-3xy-7y=0
the 3xy's go away; multiply by 2/7 to get
-(x^2+6/7)+(y^2-2y) = 0
-(x+3/7)^2+(y-1)^2 = 58/49
an hyperbola opening up and down
To integrate the given equation, we can follow the steps of line integration. Here's how you can solve it:
Step 1: Identify the coefficients and differentials of x and y.
Given equation: (3y - 7x - 3)dx + (7y - 3x - 7)dy = 0
Here, the coefficient of dx is (3y - 7x - 3), and the coefficient of dy is (7y - 3x - 7).
Step 2: Rearrange the equation.
To make the equation easier to integrate, we can rearrange it to isolate dx and dy on one side and the remaining terms on the other side.
(3y - 7x - 3)dx + (7y - 3x - 7)dy = 0
(3y - 7x - 3)dx = -(7y - 3x - 7)dy
Step 3: Divide both sides by the corresponding differentials.
Now, divide both sides of the equation by (3y - 7x - 3):
dx = -(7y - 3x - 7)dy / (3y - 7x - 3)
Step 4: Integrate both sides with respect to x and y.
Now, we can integrate both sides of the equation. Integrating dx with respect to x is relatively simple:
∫ dx = ∫ -(7y - 3x - 7)dy / (3y - 7x - 3)
x = -∫ (7y - 3x - 7)dy / (3y - 7x - 3) + C
However, integrating the right side with respect to y can be challenging due to the complicated expression. Solving this requires advanced integration techniques like partial fractions, substitution, or using software tools like Wolfram Alpha.
So, to simplify the answer, we leave the integration of the right side as it is and express the solution as:
x = ∫ -(7y - 3x - 7)dy / (3y - 7x - 3) + C
Note: If you have access to software that can perform symbolic integration, it would be more efficient to use that to obtain the exact solution.
To solve the given differential equation $\left(3y-7x-3\right)dx+\left(7y-3x-7\right)dy=0$:
Step 1: Check for Exactness
To determine whether the equation is exact, we need to test if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$, where $M$ and $N$ are the coefficients of $dx$ and $dy$ respectively.
Let's compute these partial derivatives:
$\frac{\partial M}{\partial y} = 3$ and $\frac{\partial N}{\partial x} = -3$
Since $\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$, the equation is not exact.
Step 2: Find an Integrating Factor
To make the equation exact, we need to multiply both sides of the equation by an integrating factor, denoted by $\mu$:
$\mu \left(3y-7x-3\right)dx+\mu \left(7y-3x-7\right)dy=0$
To find the integrating factor $\mu$, we use the formula $\mu = \frac{1}{\left(M \frac{\partial \mu}{\partial y}\right) - \left(N \frac{\partial \mu}{\partial x}\right)}$
First, we find the partial derivatives:
$\frac{\partial \mu}{\partial x}$ and $\frac{\partial \mu}{\partial y}$
Next, we set up the equation:
$\left(M \frac{\partial \mu}{\partial y}\right) - \left(N \frac{\partial \mu}{\partial x}\right) = (\frac{\partial \mu}{\partial x} - 3)(3) - (\frac{\partial \mu}{\partial y} - 7)(-3) = 0$
Simplify the equation:
$3\frac{\partial \mu}{\partial x} - 9 - 3\frac{\partial \mu}{\partial y} + 21 = 0$
Combine like terms:
$3\frac{\partial \mu}{\partial x} - 3\frac{\partial \mu}{\partial y} + 12 = 0$
Step 3: Solve the Integrating Factor Equation
Solve the integrating factor equation: $3\frac{\partial \mu}{\partial x} - 3\frac{\partial \mu}{\partial y} + 12 = 0$
This is a linear partial differential equation.
To solve it, assume $\mu$ is a function of $x$, and find $\frac{\partial \mu}{\partial x}$ and $\frac{\partial \mu}{\partial y}$ in terms of $x$.
$3\frac{\partial \mu}{\partial x} - 3\frac{\partial \mu}{\partial y} + 12 = 0$
Rearrange the equation:
$3\frac{\partial \mu}{\partial x} - 3\frac{\partial \mu}{\partial y} = -12$
Divide both sides by 3:
$\frac{\partial \mu}{\partial x} - \frac{\partial \mu}{\partial y} = -4$
Integrate both sides with respect to $x$:
$\int \frac{\partial \mu}{\partial x} dx - \int \frac{\partial \mu}{\partial y} dx = \int -4 dx$
$\mu + g(y) - f(x) = -4x + c$
Step 4: Simplify the Integrating Factor Equation
To simplify the integrating factor equation, we can rewrite it as:
$\mu = 4x - g(y) + f(x) + c$
Step 5: Determine the Integrating Factor
By comparing $\mu = 4x - g(y) + f(x) + c$ with the integrating factor equation, we can conclude that $\mu = 4x - y + g(y)$ is the integrating factor.
Step 6: Multiply the Equation by the Integrating Factor
Multiply both sides of the original equation by the integrating factor:
$(4x - y + g(y))\left(3y-7x-3\right)dx+(4x - y + g(y))\left(7y-3x-7\right)dy=0$
Step 7: Simplify and Solve the Equation
Expand and rearrange the equation:
$(12xy - 21x^2 - 9x - 3y^2 + 7xy + 3y + 3g(y) - 7y^2 + y + g(y))dx + (28xy - 49x^2 - 21x - 7y^2 + 21xy - 9y - 7g(y) + 49y^2 - g(y))dy = 0$
Combine like terms:
$(19xy - 35x^2 - 25x - 10y^2 + 14y + 4g(y))dx + (49xy - 70x^2 - 30x - 6g(y) + 42y^2 - 9y)dy = 0$
By matching the coefficients of $dx$ and $dy$ to zero, we can obtain a system of equations to solve for $x$, $y$, and $g(y)$:
$19xy - 35x^2 - 25x - 10y^2 + 14y + 4g(y) = 0$ ---(1)
$49xy - 70x^2 - 30x - 6g(y) + 42y^2 - 9y = 0$ ---(2)
Solve equations (1) and (2) simultaneously to find the values of $x$, $y$, and $g(y)$.
Note: Due to the complexity of these equations, the solution may require numerical methods or further simplification.