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Degree of the differential equation sin x + cos (dy/dx) = y² is
The differential equation given is: sin(x) + cos(dy/dx) = y² To know the degree of the differential equation, we have the following steps: 1. Rearrange the formula: The formula contains dy/dx that is not raised to any power directly. However, with the term cos(dy/dx) the degree calculation is compliRead more
The differential equation given is:
sin(x) + cos(dy/dx) = y²
To know the degree of the differential equation, we have the following steps:
1. Rearrange the formula: The formula contains dy/dx that is not raised to any power directly. However, with the term cos(dy/dx) the degree calculation is complicated since it deals with a trigonometric function of the derivative.
2. Degree of a differential equation: The degree of a differential equation is defined to be the highest power of the highest-order derivative after the equation has been made free from any irrational terms, fractional powers, or trigonometric functions involving derivatives.
3. This equation contains a trigonometric function, cos(dy/dx), whose argument is a first derivative, not an algebraic expression, not a polynomial, not a simple rational term. Since it’s placed within the argument of a trigonometric function, the degree is undefined.
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The general solution of rhe differential equation ydx -xdy = 0; (Given x, y > 0), is of the form
The given differential equation is: y dx - x dy = 0 This is a first-order linear differential equation. We can rewrite it as: y dx = x dy Now, divide both sides by x and y: (dx/x) = (dy/y) This is a separable differential equation, meaning we can integrate both sides separately. Integrating both sidRead more
The given differential equation is:
y dx – x dy = 0
This is a first-order linear differential equation. We can rewrite it as:
y dx = x dy
Now, divide both sides by x and y:
(dx/x) = (dy/y)
This is a separable differential equation, meaning we can integrate both sides separately.
Integrating both sides:
∫(1/x) dx = ∫(1/y) dy
The integrals of 1/x and 1/y are:
ln|x| = ln|y| + C
Now, take both sides as exponents to get rid of the logarithms:
x = C y
Therefore, the general solution is:
y = cx
where c is a constant.
So, the right answer is y = cx.
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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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