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What is the solution of the differential equation 2x3y dy + (1 - y2)(x2y2 + y2 - 1) dx = 0?
- a)x2y2 = (cx + 1)(1 - y2)
- b)x2y2 = (cx + 1)(1 + y2)
- c)x2y2 = (cx - 1)(1 - y2)
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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What is the solution of the differential equation 2x3y dy + (1 - y2)(x...
2x3y dy + (1 - y2)(x2y2 + y2 - 1) dx = 0
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Put
Solving, we get:
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What is the solution of the differential equation 2x3y dy + (1 - y2)(x...
Solution:
Given differential equation is 2x^3y dy - (1 - y^2)(x^2y^2 - y^2 + 1) dx = 0
To solve this differential equation, we can separate the variables and integrate.
Separating the variables, we get:
2x^3y dy = (1 - y^2)(x^2y^2 - y^2 + 1) dx
Now, let's integrate both sides.
∫2x^3y dy = ∫(1 - y^2)(x^2y^2 - y^2 + 1) dx
Integrating the left side with respect to y:
x^3y^2 + C1 = ∫(1 - y^2)(x^2y^2 - y^2 + 1) dx
Simplifying the right side:
x^3y^2 + C1 = ∫(x^2y^2 - x^2y^4 - y^2 + y^4 + x^2 - 1) dx
x^3y^2 + C1 = ∫(x^2 - 1 + y^4 - y^2 + x^2y^2 - x^2y^4) dx
x^3y^2 + C1 = x^3 - x + y^4x + y^2x - x^3y^4 + x^3y^2 + C2
Combining the constants:
x^3y^2 + C1 = x^3 - x + y^4x + y^2x - x^3y^4 + x^3y^2 + C2
Simplifying further, we get:
C1 = x^3 - x + y^4x + y^2x - x^3y^4 + C2
Now, we can rearrange the equation to isolate x^2y^2 terms:
C1 = x^3 - x + y^4x + y^2x - x^3y^4 + C2
C1 - C2 = x^3 - x + y^4x + y^2x - x^3y^4
C1 - C2 = x^3 + x(y^4 + y^2 - 1) - x^3y^4
Now, we can rewrite the equation in the form of x^2y^2:
x^2y^2 = (C1 - C2)/(x^3 + x(y^4 + y^2 - 1) - x^3y^4)
Therefore, the solution to the given differential equation is:
x^2y^2 = (C1 - C2)/(x^3 + x(y^4 + y^2 - 1) - x^3y^4)
So, the correct answer is option C) x^2y^2 = (cx - 1)(1 - y^2).
Given differential equation is 2x^3y dy - (1 - y^2)(x^2y^2 - y^2 + 1) dx = 0
To solve this differential equation, we can separate the variables and integrate.
Separating the variables, we get:
2x^3y dy = (1 - y^2)(x^2y^2 - y^2 + 1) dx
Now, let's integrate both sides.
∫2x^3y dy = ∫(1 - y^2)(x^2y^2 - y^2 + 1) dx
Integrating the left side with respect to y:
x^3y^2 + C1 = ∫(1 - y^2)(x^2y^2 - y^2 + 1) dx
Simplifying the right side:
x^3y^2 + C1 = ∫(x^2y^2 - x^2y^4 - y^2 + y^4 + x^2 - 1) dx
x^3y^2 + C1 = ∫(x^2 - 1 + y^4 - y^2 + x^2y^2 - x^2y^4) dx
x^3y^2 + C1 = x^3 - x + y^4x + y^2x - x^3y^4 + x^3y^2 + C2
Combining the constants:
x^3y^2 + C1 = x^3 - x + y^4x + y^2x - x^3y^4 + x^3y^2 + C2
Simplifying further, we get:
C1 = x^3 - x + y^4x + y^2x - x^3y^4 + C2
Now, we can rearrange the equation to isolate x^2y^2 terms:
C1 = x^3 - x + y^4x + y^2x - x^3y^4 + C2
C1 - C2 = x^3 - x + y^4x + y^2x - x^3y^4
C1 - C2 = x^3 + x(y^4 + y^2 - 1) - x^3y^4
Now, we can rewrite the equation in the form of x^2y^2:
x^2y^2 = (C1 - C2)/(x^3 + x(y^4 + y^2 - 1) - x^3y^4)
Therefore, the solution to the given differential equation is:
x^2y^2 = (C1 - C2)/(x^3 + x(y^4 + y^2 - 1) - x^3y^4)
So, the correct answer is option C) x^2y^2 = (cx - 1)(1 - y^2).
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