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If the function x, x2,, x3 are independent then the equation of differential equation formed with these independent solution x3y”’ – 3x2y” + 6xy’ – 6y equals to_______
Correct answer is '0'. Can you explain this answer?
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If the function x, x2,, x3 are independent then the equation of differ...
Understanding the Given Information
The given information states that the functions x, x^2, and x^3 are independent. This means that none of these functions can be written as a linear combination of the others. In other words, there is no constant value such that one of these functions can be obtained by multiplying another function by that constant.
Forming the Differential Equation
To form the differential equation using the given solutions, we can start by assuming that the dependent variable is y. Then, we can express the given solutions in terms of y as follows:
x^3y = y^4
3x^2y = 3xy^3
6xy = 6xy
6y = 6y
Combining the Expressions
Next, we can substitute these expressions into the given equation:
y^4 + 3xy^3 + 6xy + 6y = 0
Simplifying the Equation
To simplify the equation further, we can factor out a common factor of y:
y(y^3 + 3xy^2 + 6x + 6) = 0
Conclusion
From the simplified equation, we can see that there are two possible solutions for y: y = 0 and y^3 + 3xy^2 + 6x + 6 = 0. However, since the given information states that x, x^2, and x^3 are independent solutions, the only valid solution is y = 0.
Therefore, the correct answer to the differential equation formed with the independent solutions x, x^2, and x^3 is 0.
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If the function x, x2,, x3 are independent then the equation of differential equation formed with these independent solution x3y”’ – 3x2y” + 6xy’ – 6y equals to_______Correct answer is '0'. Can you explain this answer?
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