Quiz I D. E.

A linear differential equation is a type of differential equation where the dependent variable and its derivatives appear in a linear manner. In other words, the equation can be expressed as a linear combination of the dependent variable and its derivatives, with constant coefficients. This type of equation is characterized by its linearity, which allows for the use of various mathematical techniques and methods to solve it. Therefore, the given answer "Linear differential equation" is correct as it accurately describes the type of equation being referred to in the question.

The given differential equation is in the form of dr/dθ = 500θn - r/θ. To solve this equation, we can use the method of integrating factors. The integrating factor is given by θ. By multiplying both sides of the equation by θ, we get θdr/dθ = 500θ^2n - r. This can be rearranged as d(θr)/dθ = 500θ^2n. Integrating both sides with respect to θ gives us θr = 500θ^(2n+1)/(2n+1) + C. Simplifying further, we get r = 500θ^(2n+1)/(2n+1θ) + C/θ. Therefore, the answer is θ.

The order of a differential equation is determined by the highest derivative present in the equation. In this case, the given differential equation does not have any derivatives, so the order is 0. However, the options provided do not include 0 as a possible answer. The closest option is 1/2, but this is not a valid order for the given equation. Therefore, the correct answer is not available.

The given differential equation is a separable equation, which means it can be written in the form dy/dx = f(x)g(y). In this case, we have y dy/dx = x - 1, which can be rearranged as y dy = (x - 1) dx. Integrating both sides, we get (1/2)y^2 = (1/2)x^2 - x + C, where C is the constant of integration. Since y(1) = 1, we can substitute this into the equation to solve for C. Plugging in the values, we get (1/2)(1)^2 = (1/2)(1)^2 - 1 + C, which simplifies to 1/2 = -1/2 + C. Solving for C, we find C = 1. Substituting this back into the equation, we get (1/2)y^2 = (1/2)x^2 - x + 1, which simplifies to y^2 = x^2 - 2x + 2.

The given differential equation is separable, meaning it can be written as dy/(y^2)^(1/5) = 60dx. Integrating both sides gives 5/4(y^2)^(4/5) = 60x + C, where C is the constant of integration. Solving for y gives y^2 = (4/5)(60x + C)^(5/4). Since y(0) = 0, we can substitute this into the equation to find C = 0. Therefore, y^2 = (4/5)(60x)^(5/4), which has two possible solutions: y = ±(2/5)(60x)^(5/8). Hence, the correct answer is two solutions.

The question is asking for the derivative of the function x^3y with respect to x. To find this derivative, we use the product rule of differentiation. The derivative of x^3y with respect to x is equal to the derivative of x^3 times y plus x^3 times the derivative of y with respect to x. Therefore, the correct answer is 3x^2y + x^3(dy/dx).

The degree of a differential equation is determined by the highest power of the derivative present in the equation. In this case, since there is no derivative present, the degree is 0. Therefore, the correct answer is not available.

The given differential equation exdy/dx+3y=x2y can be separated into two variables, x and y, by moving the terms involving y to one side and the terms involving x to the other side. This separation allows us to solve the equation by integrating both sides with respect to their respective variables. Additionally, the equation is linear because the highest power of y and its derivatives is 1, and the highest power of x is also 1. Therefore, the correct answer is both separable and linear.

The equation is reducible to separable form if it can be written in the form of dy/dx = f(x)g(y), where f(x) is a function of x only and g(y) is a function of y only. This allows us to separate the variables and integrate both sides of the equation separately.

The given equation is a first-order linear homogeneous differential equation. To solve it, we can use the method of separation of variables. By rearranging the equation, we get dy/dx = -tan y tan x + cos x sec y.

We can rewrite the equation as dy/(tan y sec y) = (cos x - tan x) dx.

Integrating both sides, we get ∫dy/(tan y sec y) = ∫(cos x - tan x) dx.

The integral of dy/(tan y sec y) can be simplified to ∫cos y dy. And the integral of (cos x - tan x) dx can be simplified to sin x - ln|cos x|.

Therefore, we have sin y = x + c - ln|cos x|. Rearranging the equation, we get sin y = (x + c) cos x.

The given differential equation is not exact, so we need an integrating factor to make it exact. An integrating factor is a function that, when multiplied to the equation, makes the left-hand side equal to the derivative of a product. In this case, if we multiply the equation by x/y, the left-hand side becomes (x^2/y)dy - (xy/y)dx = xdx - ydy, which is the derivative of xy. Therefore, x/y is an integrating factor for the given equation.

The given differential equation is a first-order linear ordinary differential equation. To solve it, we can use the method of integrating factors. By multiplying the entire equation by the integrating factor, which is e^∫φ(x) dx, we can transform the equation into a form that can be easily integrated. After integrating both sides, we obtain the solution y = φ(x) - 1 + c e^(-φ(x)), where c is the constant of integration.

The given equation is a first-order linear differential equation. To solve it, we can rearrange it to the form dy/dx + P(x)y = Q(x), where P(x) = -x(3/4) and Q(x) = x(3/4)y^3. The integrating factor (I.F.) for this equation is e^(∫P(x)dx). In this case, the I.F. is e^(∫-x(3/4)dx) = e^(-3x/4). Multiplying both sides of the equation by the I.F., we get e^(-3x/4) * y4 dx = e^(-3x/4) * (x(3/4)-y^3x)dy. This can be rewritten as d(e^(-3x/4) * y^4) = e^(-3x/4) * x(3/4)dy. Integrating both sides, we obtain e^(-3x/4) * y^4 = ∫e^(-3x/4) * x(3/4)dy. Solving for y, we find y = (e^(3x/4) * ∫x(3/4)dy)^(1/4), which simplifies to y = x(7/4). Therefore, the correct answer is y(7/4).

The given differential equation xy' = y log y represents the family of curves y = e^(500x). This can be determined by rearranging the equation as y'/y = log y/x and integrating both sides. The left side becomes ln|y| and the right side becomes log y. Solving for y gives y = e^(500x), which is the equation of the family of curves.

The given equation is a first-order nonlinear ordinary differential equation. To solve it, we can use the method of separation of variables. By rearranging the equation, we have dy/tan(y) = sin(x+y) + sin(x-y) dx. Integrating both sides, we get ln|sec(y)| = -cos(x+y) - cos(x-y) + K, where K is the constant of integration. Exponentiating both sides, we have |sec(y)| = e^(-cos(x+y) - cos(x-y) + K). Since the absolute value of sec(y) is always positive, we can remove the absolute value sign. Thus, sec(y) = e^(-cos(x+y) - cos(x-y) + K). Simplifying further, we have sec(y) = e^(2cos(x) + K). Rearranging the equation, we get sec(y) + 2cos(x) = C, where C = e^K. Therefore, the correct answer is sec ⁡y+2 cos⁡x=C.

The integrating factor of a differential equation is a function that is multiplied to both sides of the equation in order to make it easier to solve. In this case, the given differential equation is in the form of (xy^3+y)dx+2(x^2y^2+x+y^4)dy = 0. To find the integrating factor, we can look for a function that depends only on y. The correct answer, 1/y, is a function that only depends on y. Multiplying both sides of the equation by 1/y will help to simplify the equation and make it easier to solve.

This is a first-order linear ordinary differential equation. To solve it, we can use the method of integrating factors. The integrating factor is e^(∫3dx) = e^(3x). Multiplying both sides of the equation by the integrating factor, we get e^(3x)(2dy/dx + 3y) = e^(2x). This can be rewritten as d/dx(e^(3x)y) = e^(2x). Integrating both sides with respect to x, we get e^(3x)y = (1/2)e^(2x) + C. Solving for y, we have y = (1/2)e^(-x) + Ce^(-3x). Using the initial condition y(0) = 5, we can substitute the values and find the constants. The solution is y = (1/2)e^(-x) + 4e^(-3x). Therefore, the correct answer is A e^(-1.5x) + Bex.

The given equation is a first-order linear ordinary differential equation. To solve it, we can use the method of integrating factors. The integrating factor for this equation is e^(∫sinz dz) = e^(-cosz). Multiplying both sides of the equation by this integrating factor, we get x*sinz*e^(-cosz)dz + (x^3 - 2x^2*cosz + cosz)*e^(-cosz)dx = 0. This can be simplified to d(x*e^(-cosz)) = 0. Integrating both sides, we get x*e^(-cosz) = C, where C is the constant of integration. Dividing both sides by x gives us the final answer, which is ex^2 / x.

The given differential equation is a first-order linear homogeneous equation. To solve it, we can use the method of integrating factors. By multiplying the entire equation by the integrating factor, which is e^(x^2), we can convert the left side into the derivative of (y*e^(x^2)). Integrating both sides and solving for y, we obtain the solution Y=1/(1+cex^2), where c is the constant of integration.

The given differential equation is a first-order linear homogeneous equation. To solve it, we can use the method of separable variables. By rearranging the equation, we get (x-y^2)dx + 2xydy = 0. Dividing both sides by x(x-y^2), we obtain dx/(x-y^2) + 2ydy/x = 0. Integrating both sides, we get ln|x-y^2| + 2ln|x| = ln|c|, where c is the constant of integration. Simplifying this equation, we have ln|x(x-y^2)^2| = ln|c|. Taking the exponential of both sides, we get x(x-y^2)^2 = c. Rearranging this equation, we obtain xe(y^2/x) = c. Therefore, the correct answer is xe(y^2/x) = c.