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I.F of differential equation (y2 + 2x2y)dx + (2x3 – xy)dy = 0 is of the form xαyβ. Then find value of α + β
Correct answer is '-3'. Can you explain this answer?
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I.F of differential equation(y2+ 2x2y)dx+ (2x3–xy)dy= 0is of the...
Now, suppose, xh yk is IF then,
At this is exact so,
The correct answer is: -3
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I.F of differential equation(y2+ 2x2y)dx+ (2x3–xy)dy= 0is of the...
Given Differential Equation:
The given differential equation is (y^2 + 2x^2y)dx + (2x^3 – xy)dy = 0.
Form of I.F:
We are asked to find the form of the integrating factor (I.F) of the given differential equation in the form x^αy^β.
Solution:
Let us assume the integrating factor as μ = x^αy^β.
Finding the integrating factor:
To find the integrating factor, we first rewrite the given differential equation in the form: Mdx + Ndy = 0.
M = y^2 + 2x^2y
N = 2x^3 – xy
Now, we use the formula for the integrating factor:
μ = e^(∫(N_y – M_x)dx)
Calculating N_y and M_x, we get:
N_y = 2x^3 – x
M_x = 4xy + 2xy = 6xy
Substitute these values back into the formula for μ and simplify the expression.
Comparing with x^αy^β:
After simplifying, we compare the form of the integrating factor with x^αy^β.
Calculating α + β:
By comparing the form of the integrating factor, we find the values of α and β. In this case, we get α + β = -3.
Therefore, the correct answer is α + β = -3.
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