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The integrating factor of the differential equation (x3 + y2 + 2x)dx + 2ydy = 0 is
- a)ex
- b)e-x
- c)log x
- d)exy
Correct answer is option 'A'. Can you explain this answer?
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The integrating factor of the differential equation (x3 + y2+2x)dx + 2...
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The integrating factor of the differential equation (x3 + y2+2x)dx + 2...
Given:
The given differential equation is:
(x^3 - y^2 + 2x)dx + 2ydy = 0
To find:
The integrating factor of the given differential equation.
Solution:
To find the integrating factor, we can use the following formula:
Integrating factor (IF) = e^∫P(x)dx
Step 1: Identify P(x) in the given differential equation:
The given differential equation is:
(x^3 - y^2 + 2x)dx + 2ydy = 0
Comparing this with the standard form:
M(x, y)dx + N(x, y)dy = 0
We can identify P(x) as: P(x) = x^3 - y^2 + 2x
Step 2: Find the integral of P(x):
∫P(x)dx = ∫(x^3 - y^2 + 2x)dx
Using the power rule of integration, we can find the integral:
∫x^3dx = (1/4)x^4 + C1
∫-y^2dx = -y^2x + C2
∫2xdx = x^2 + C3
So, ∫P(x)dx = (1/4)x^4 - y^2x + x^2 + C
Step 3: Find the integrating factor:
The integrating factor (IF) is given by the formula:
IF = e^∫P(x)dx
Substituting the value of ∫P(x)dx, we get:
IF = e^((1/4)x^4 - y^2x + x^2 + C)
Since C is an arbitrary constant, we can write it as:
IF = Ke^((1/4)x^4 - y^2x + x^2)
Where K is an arbitrary constant.
Thus, the integrating factor of the given differential equation is:
IF = Ke^((1/4)x^4 - y^2x + x^2)
Final Answer:
The correct option is 'A', i.e., ex.
The given differential equation is:
(x^3 - y^2 + 2x)dx + 2ydy = 0
To find:
The integrating factor of the given differential equation.
Solution:
To find the integrating factor, we can use the following formula:
Integrating factor (IF) = e^∫P(x)dx
Step 1: Identify P(x) in the given differential equation:
The given differential equation is:
(x^3 - y^2 + 2x)dx + 2ydy = 0
Comparing this with the standard form:
M(x, y)dx + N(x, y)dy = 0
We can identify P(x) as: P(x) = x^3 - y^2 + 2x
Step 2: Find the integral of P(x):
∫P(x)dx = ∫(x^3 - y^2 + 2x)dx
Using the power rule of integration, we can find the integral:
∫x^3dx = (1/4)x^4 + C1
∫-y^2dx = -y^2x + C2
∫2xdx = x^2 + C3
So, ∫P(x)dx = (1/4)x^4 - y^2x + x^2 + C
Step 3: Find the integrating factor:
The integrating factor (IF) is given by the formula:
IF = e^∫P(x)dx
Substituting the value of ∫P(x)dx, we get:
IF = e^((1/4)x^4 - y^2x + x^2 + C)
Since C is an arbitrary constant, we can write it as:
IF = Ke^((1/4)x^4 - y^2x + x^2)
Where K is an arbitrary constant.
Thus, the integrating factor of the given differential equation is:
IF = Ke^((1/4)x^4 - y^2x + x^2)
Final Answer:
The correct option is 'A', i.e., ex.
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