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The sum of the order and the degree of the differential equation d/dx ((dy/dx)³) is
The given differential equation is as follows: d/dx ((dy/dx)³) We will break it down step by step to find the order and degree of the differential equation. Order: The order of a differential equation is the highest derivative of the unknown function in the equation, which in this case is y. In theRead more
The given differential equation is as follows:
d/dx ((dy/dx)³)
We will break it down step by step to find the order and degree of the differential equation.
Order: The order of a differential equation is the highest derivative of the unknown function in the equation, which in this case is y.
In the expression d/dx ((dy/dx)³), we differ_intiate the third power of the first derivative, dy/dx. Therefore, the highest order in the expression is the second derivative of y, that is, d²y/dx² .
Thus, the order is 2 .
Degree: The degree of a differential equation is the exponent of the highest-order derivative, provided the equation is polynomial in the derivatives.
Here the highest-order derivative is (dy/dx)² (occurs after differentiation), the degree of the equation is 1 because the expression (dy/dx)³ is raised to the first power (after differentiation).
Hence, the degree is 1.
Sum of the order and degree: The sum of the order and the degree is:
Order + Degree = 2 + 1 = 3
Hence, the answer is 3.
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The order and degree of the differential equation d²y/dx² + (dy/dx)¹/⁴ + x¹/⁵ = 0 are respectively
The differential equation given is: d²y/dx² + (dy/dx)^(1/4) + x^(1/5) = 0 Order: The order of a differential equation is the highest derivative of the unknown function, which, in this case, is y. The highest derivative of the given equation is d²y/dx², the second derivative of y with respect to x. SRead more
The differential equation given is:
d²y/dx² + (dy/dx)^(1/4) + x^(1/5) = 0
Order: The order of a differential equation is the highest derivative of the unknown function, which, in this case, is y.
The highest derivative of the given equation is d²y/dx², the second derivative of y with respect to x. So, the order is 2.
Degree: The degree of a differential equation is the exponent of the highest-order derivative, provided the equation is polynomial in the derivatives.
To find the degree, we remove the fractional exponent in the term (dy/dx)^(1/4). We will multiply both sides of the equation by 4 to remove the fractional power. The equation becomes a polynomial in derivatives after that, and the highest derivative is d²y/dx², which has an exponent of 2.
Therefore, the degree is 4.
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Solution of the differential equation 2xdy – ydx = 0 represents:
The given differential equation is: 2x dy - y dx = 0 Step 1: Rearrange the equation Put the equation into the separable variable form as: (2x dy) = (y dx) Now dividing both sides by x and changing, dy/dx = y / (2x) Step 2: Solution through Separating Variables Write it, dy/y = dx/(2x) InteRead more
The given differential equation is:
2x dy – y dx = 0
Step 1: Rearrange the equation
Put the equation into the separable variable form as:
(2x dy) = (y dx)
Now dividing both sides by x and changing,
dy/dx = y / (2x)
Step 2: Solution through Separating Variables
Write it,
dy/y = dx/(2x)
Integrating on both sides:
∫ (1/y) dy = ∫ (1/2x) dx
ln|y| = (1/2) ln|x| + C
Step 3: Putting into Exponential Form
Now, taking the exponent on both sides:
y = e^C * x^(1/2)
Let e^C = C’, then:
y = C’ x^(1/2)
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If |A| = |kA|, where A is a square matrix of order 2, then sum of all possible values of k is
We are given that A is a square matrix of order 2, and |A| = |kA|. We have to compute the sum of all possible values of k. First recall that if A is any square matrix of order 2, then the determinant of kA where k is any scalar is given by |kA| = k² |A| This is because for a 2x2 matrix, multiplicatiRead more
We are given that A is a square matrix of order 2, and |A| = |kA|. We have to compute the sum of all possible values of k.
First recall that if A is any square matrix of order 2, then the determinant of kA where k is any scalar is given by
|kA| = k² |A|
This is because for a 2×2 matrix, multiplication of the matrix A by scalar k scales the determinant by k².
We know that |A| = |kA|. Using the above formula, we substitute for the value of |kA|,
|A| = k² |A|
If |A| ≠ 0, we can divide both sides by |A| to get,
1 = k²
This gives two values for k
k = 1 or k = -1
Therefore, the sum of all possible values of k is:
1 + (-1) = 0
So, the correct answer is 0.
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If for a square matrix A, A² – A + I = O, then A⁻¹ equals
We have the equation of a square matrix A as A² - A + I = O Rearrange the equation to get A² - A = -I Now we factor the left-hand side, A(A - I) = -I To get the inverse of A (A⁻¹), multiply both sides of the equation by A⁻¹, A⁻¹ * A(A - I) = A⁻¹ * (-I) This gives us, (A - I) = -A⁻¹ Thus we can writeRead more
We have the equation of a square matrix A as
A² – A + I = O
Rearrange the equation to get
A² – A = -I
Now we factor the left-hand side,
A(A – I) = -I
To get the inverse of A (A⁻¹), multiply both sides of the equation by A⁻¹,
A⁻¹ * A(A – I) = A⁻¹ * (-I)
This gives us,
(A – I) = -A⁻¹
Thus we can write
A⁻¹ = I – A
Hence, the correct answer is option (c) I – A.
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