Question

For each of the following D.E find the general solution. 14) (3y−7x+7)dx−(3x−7y−3)dy=0 15) (x+y−2)dx+(2x+2y+4)dy=0

Answer 1

To find the general solution for each of the given differential equations, rearrange the equations to separate the variables, integrate both sides with respect to their respective variables, and add a constant of integration. Then, simplify the equation to obtain the general solution.

Step-by-step explanation:

To find the general solution for each of the given differential equations:

14) (3y−7x+7)dx−(3x−7y−3)dy=0:
To begin, we can rearrange the equation to separate the variables:
(3y-7x+7)dx=(3x-7y+3)dy

Now, integrate both sides with respect to their respective variables and add a constant of integration:
∫(3y-7x+7)dx = ∫(3x-7y+3)dy
Integrating, we get:
3xy - 7x^2/2 + 7x + k1 = 3xy - 7y^2/2 + 3y + k2
Simplifying, we have the general solution:
7x^2 - 7y^2 + 6y - 7x + k = 0, where k is the constant of integration.

15) (x+y−2)dx+(2x+2y+4)dy=0:
Similarly, rearrange the equation to separate the variables:
(x+y-2)dx = -(2x+2y+4)dy

Integrating both sides and adding a constant of integration:
∫(x+y-2)dx = -∫(2x+2y+4)dy
Integrating, we get:
x^2/2 + xy - 2x + k1 = -2xy - 2y^2/2 - 4y + k2
Simplifying, we have the general solution:
x^2 + 2xy + 4y^2 + 4x - 2y + k = 0, where k is the constant of integration.

Answer 2

Final Answers:

14) (3y-7x+7)dx - (3x-7y-3)dy = 0

The general solution is: y^2 - xy + 7x = C, where C is an arbitrary constant.

15) (x+y-2)dx + (2x+2y+4)dy = 0

The general solution is: (x^2 + xy - 2y) = Cx, where C is an arbitrary constant.

Step-by-step explanation:

Both equations are in the form of homogeneous differential equations, which can be solved by separating the variables and integrating both sides.

14) (3y-7x+7)dx - (3x-7y-3)dy = 0:

Rearrange the equation to separate variables:

(3y+7)dy = (7x-3)dx

Integrate both sides:

3y^2 + 7y = 7x^2 - 3x + C1

Combine terms and simplify:

y^2 - xy + 7x = C (where C = C1/3)

15) (x+y-2)dx + (2x+2y+4)dy = 0:

Rearrange the equation to separate variables:

(y-2)dy = -(2x+4)dx

Integrate both sides:

y^2 - 2y = -2x^2 - 4x + C2

Combine terms and simplify:

(x^2 + xy - 2y) = Cx (where C = -C2/2)

Therefore, these are the general solutions for each differential equation. Remember that the arbitrary constants C1 and C2 can be combined into a single constant C for each solution.