Quiz I D. E.
Questions and Answers
- 1.
The solution of differential equation (x-y2) dx +2xy dy=0 is given by
- A.
xe(y2/x)=c
- B.
Ye(y2/x)=c
- C.
Xe(x2/y)=c
- D.
Ye(x2/y)=c
Correct Answer
A. xe(y2/x)=cExplanation
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.Rate this question:
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- 2.
The solution of dy/dx tan y = sin(x+y)+sin(x-y) is given by
- A.
sec y+2 cosx=C
- B.
Sec2y-2 cosx=C
- C.
Secy-2 cosx=C
- D.
Sec2y+2 cosx=C
Correct Answer
A. sec y+2 cosx=CExplanation
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 cosx=C.Rate this question:
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- 3.
The Integrating Factor of the D.E. (xy3+y)dx+2(x2y2+x+y4)dy =0, is given by
- A.
1/y
- B.
y
- C.
-1/y
- D.
-y
Correct Answer
A. 1/yExplanation
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.Rate this question:
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- 4.
The I. F.of the Bernoulli’s equation x sinz dz+(x3 - 2x2 cosz + cosz)dx=0 is
- A.
Ex2 /x
- B.
(-x)/ex2
- C.
E-x2/x
- D.
X/ex2
Correct Answer
A. Ex2 /xExplanation
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.Rate this question:
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- 5.
The order of the given differential equation is
- A.
4
- B.
2
- C.
3/2
- D.
1/2
Correct Answer
A. 4Explanation
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.Rate this question:
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- 6.
The I.F. of the equation y4 dx = (x(3/4)-y3x)dy is
- A.
Y(7/4)
- B.
X(7/4)
- C.
Y(7/4)
- D.
Y((-7)/4)
Correct Answer
A. Y(7/4)Explanation
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).Rate this question:
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- 7.
The solution of (dy/dx) + y (dφ/dx) = φ (x) (dφ/dx) is
- A.
Y= φ(x)-1+ c e-φ
- B.
Y = [φ(x)-1] e-φ + c
- C.
Y = x φ(x)-c e-φ
- D.
Y = c eφ
Correct Answer
A. Y= &pHi;(x)-1+ c e-&pHi;Explanation
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.Rate this question:
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- 8.
The differential equation dy/dx=60(y2)1/5 , x > 0, y(0)=0 has
- A.
Two solutions
- B.
No solution
- C.
Infinite number of solutions
- D.
A unique solution
Correct Answer
A. Two solutionsExplanation
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.Rate this question:
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- 9.
The solution of (dy/dx) + 2xy = 2xy2 , is
- A.
Y=1/(1+cex2 )
- B.
Y=cx/(1+ex2))
- C.
Y=1/(1-cex)
- D.
Y=cx/(1+e-x2 )
Correct Answer
A. Y=1/(1+cex2 )Explanation
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.Rate this question:
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- 10.
The degree of given differential equation is
- A.
2
- B.
1
- C.
3
- D.
3/2
Correct Answer
A. 2Explanation
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.Rate this question:
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- 11.
Which of the following is not an integrating factor of x dy – y dx =0
- A.
X/y
- B.
1/xy
- C.
1/(x2 + y2)
- D.
1/x2
Correct Answer
A. X/yExplanation
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.Rate this question:
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- 12.
The equation reducible to separable form if
- A.
![Quiz I D. E. - Quiz]()
- B.
![Quiz I D. E. - Quiz]()
- C.
![Quiz I D. E. - Quiz]()
- D.
![Quiz I D. E. - Quiz]()
Correct Answer
A.Explanation
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.Rate this question:
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- 13.
is called
- A.
Linear differential equation
- B.
Second order differential
- C.
Homogeneous differential equation
- D.
Bernoulli’s equation
Correct Answer
A. Linear differential equationExplanation
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.Rate this question:
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- 14.
The solution of the differential equation y dy/dx=x-1 satisfying y(1)=1 is:
- A.
Y2 =x2 -2x+2
- B.
Y =2x2 -x-1
- C.
Y =x2 -2x+2
- D.
Y2 =2x2 -x-1
Correct Answer
A. Y2 =x2 -2x+2Explanation
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.Rate this question:
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- 15.
The general solution of dy/dx +tan y tan x =cos x sec y is
- A.
Sin y = (x + c) cos x
- B.
Cos y = (x + c) sin x
- C.
Sec y = (x + c)
- D.
2 sin y = (x + c –sin x cos x) sec x
Correct Answer
A. Sin y = (x + c) cos xExplanation
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.Rate this question:
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- 16.
2dy/dx+3y=e-x, y(0)=5
- A.
A e-1.5x+Bex
- B.
A e-1.5x+Bxex
- C.
A e1.5x+Bex
- D.
A e-1.5x+Bxex
Correct Answer
A. A e-1.5x+BexExplanation
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.Rate this question:
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- 17.
Integrating factor of dr/dθ= 500θn- r/θ
- A.
θ
- B.
2θ
- C.
3θ
- D.
4θ
Correct Answer
A. θExplanation
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.Rate this question:
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- 18.
The differential Equations regarding the family of the curve y=e500x
- A.
Xy’=y log y
- B.
Xy’=x log x
- C.
Yy’=y log y
- D.
X+y’=y log y
Correct Answer
A. Xy’=y log yExplanation
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.Rate this question:
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- 19.
The differential equation exdy/dx+3y=x2y is
- A.
Both separable and linear
- B.
Separable and not linear
- C.
Linear and not separable
- D.
Neither separable nor linear
Correct Answer
A. Both separable and linearExplanation
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.Rate this question:
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- 20.
Suppose y is a function of x. Which of the following is d(x3y)/dx
- A.
3x2y+x3dy/dx
- B.
3x2y
- C.
3x2dy/dx
- D.
3x2y+x3
Correct Answer
A. 3x2y+x3dy/dxExplanation
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).Rate this question:
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Mar 20, 2023Quiz Edited by
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Jan 23, 2018Quiz Created by
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