Sheet1Differential Equation

QUESTIONSCHOICESANSWERDISCUSSION1It is an equation involving derivatives or differentialsa. Differential Equationb. Quadratic Equationc. Cubic Equationd. Biquadratic Equationa. Differential EquationA Differential Equation is one which contains within at least one derivative or it is an equation involving derivatives or differentials. Sometimes, for analytical convenience, the differential equation is written in terms of differentials.

2Type of solution to a differential equation in which it does not contain any arbitrary constants.a. General Solutionb. Particular Solutionc. Actual solutiond. None of the aboveb. General Solution There are two types of solution to an ordinary differential equation. The first is the Particular Solution in which it does not contain any arbitrary constant. The 2nd is the General Solution in which it contains at least one arbitrary constant.

3What is called to the largest power or exponent of the highest-ordered derivative present in the equation of a differential equation?a. Orderb. Degreec. Based. Typeb. DegreeThe degree of a differential equation is the largest power or exponent.

4Values of the solution and/or its derivative(s) at specific pointsa. Steady stateb. Initial conditionsc. Homogenousd. Integrating Factorb. Initial conditionsAn initial value problem is an ordinary differential equation together with specified value, called the initial condition, of the unknown function at a given point in the domain of the solution

5Is the most general form that the solution can take and doesnt take any initial conditions into account.a. Particular Solutionb. Actual Solutionc. General Solutiond. Singular Solutionc. General solutionA weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives appearing in the equation may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense.

6Is a differential equation along with an appropriate number of initial conditions.

a. Initial Value Problemb. Initial Conditionc. General Equationd. None of the abovea. Initial Value ProblemAn Initial Value Problem (or IVP) is a differential equation along with an appropriate number of initial conditions.

7It is the specific solution that not only satisfies the differential equation, but also satisfies the given initial condition(s).

a. Actual Solutionb. Particular Solution c. General Solutiond. Singular Solutiona. Actual SolutionThe actual solution to a differential equation is the specific solution that not only satisfies the differential equation, but also satisfies the given initial condition(s).

8A differential equation that can be writtenin the form y' = F(y/x). It is said to be scaleable. Bysubstituting for y/x the equation may be made separable.a. Homogeneous b. Non Homogenousc. Lineard. Non lineara. HomogeneousHomogeneous is a differential equation that can be written in the form y' = F(y/x). It is said to be scaleable. By substituting for y/x the equation may be made separable.

9A function by which a differential equation is multiplied so that each side may be recognized as a derivative (and then be integrated).a. First Degreeb. Variablec. Derivatived. Integrating factord. Integrating factorIntegrating factor is a function by which a differential equation is multiplied so that each side may be recognized as a derivative (and then be integrated)

10A first-order differential equation that can bewritten in the form dy/dx + P(x)y = Q(x)yn. If n = 0 or 1then the equation is lineara. Bernoullib. Exact c. Lineard. Homogenousa. BernoulliBernoulli is a first-order differential equation that can be written in the form dy/dx + P(x)y = Q(x)yn. If n = 0 or 1then the equation is linear.

11A differential equation in which the dependentand independent variables can be algebraically separated on opposite sides of the equation.a. Separableb. Homogenousc. Non homogenousd. Exacta. SeparableA differential equation in which the dependent and independent variables can be algebraically separated onopposite sides of the equation is called separable.

12Is a particular solution which cannot be found bysubstituting a value for C. There may be one or severalsingular solutions for a differential equation.a. Singular solutionb. General solutionc. Particular solutiond. Actual solutionc. Particular solutionA singular solution of a differentialequation is a particular solution which cannot be found by substituting a value for C. There may be one or several singular solutions for a differential equation.

13A curve which is approached but notcrossed by the plot. Same as asymptotic curvea. Boundaryb. Orthogonalc. Trajectoryd. None of the abovea. BoundaryBoundary curve is a curve which is approached but not crossed by the plot. Same as asymptotic curve

14It describe the relations between the dependent and independent variables. An equal sign "=" is required in every equation. a. Variable separableb. Arbitraryc. Equationsd. All of the above c. EquationsAn equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent).

15Differential equations that involve only ONE independent variable. a. Ordinaryb. Generalc. Particulard. Partiala. OrdinaryAn ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable.

16Who invented Differential Equation?a. John Bernoullib. Isaac Newtonc. Eugene Torred. Joel Lassa. PartialIsaac Newton's invention invention of what we now call Differential Equations has remained a cornerstone of pure & applied science and the subject is still generating new ideas.

17It is the highest derivative that appears in the differential equation. a. Degreeb. Orderc. Partial derivatived. None of the abovea. OrderThe order of a differential equation is the highest derivative that appears in the differential equation.

18What does a differential equation called if there are no multiplications among dependent variables and their derivatives. In other words, all coefficients are functions of independent variables. a. Bernoullib. Linearc. Partiald. Exactb. LinearA differential equation is called linear if there are no multiplications among dependent variables and their derivatives. In other words, all coefficients are functions of independent variables.

19For a non-linear differential equation, if there are no multiplications among all dependent variables and their derivatives in the highest derivative term, the differential equation is considered to be what? a. Exactb. Non linearc. Lineard. Quasi-lineard. Quasi-linearFor a non-linear differential equation, if there are no multiplications among all dependent variables and their derivatives in the highest derivative term, the differential equation is considered to be quasi-linear.

20Solutions obtained from integrating the differential equations are called what?a. Exactb.Generalc. Lineard. Quasi-linear b. General solutions Solutions obtained from integrating the differential equations are called general solutions. The general solution of a nth order ordinary differential equation contains n arbitrary constants resulting from integrating n times.

21Solutions obtained by assigning specific values to the arbitrary constants in the general solutionsa. Actualb. General c. Particulard. Singularc. Particular solutionsParticular solutions are the solutions obtained by assigning specific values to the arbitrary constants in the general solutions.

22Solutions that can not be expressed by the general solutions.a. Singular b. Particualrc. Generald. Actuala. Singular solutions Solutions that can not be expressed by the general solutions are called singular solutions.

23Constrains that are specified at the initial point, generally time point, are called what? a. Initial conditionsb. Initial value c. both a & bd. None of the abovea. Initial conditionsConstrains that are specified at the initial point, generally time point, are called initial conditions. Problems with specified initial conditions are called initial value problems.

24Constrains that are specified at the boundary points, generally space points, are called what conditions? a. Abitraryb. Initial conditonsc. Boundary conditionsd. single conditionc. Boundary conditionsConstrains that are specified at the boundary points, generally space points, are called boundary conditions. Problems with specified boundary conditions are called boundary value problems.

25A differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous timesa. SDEb. DDEc. DAEd. SEDb. DDEIn general, delay differential equations ex-hibit much more complicated dynamics than ordinary differential equations sincea time delay could cause a stable equilibrium to become unstable and cause thepopulations to fluctuate.

26A differential equation in which one or more of the terms is a stochastic process.a. DDE b. DAEc. SDEd. SEDC. SDE Stochastic differential equations (SDEs) are, conceptually, ones where the the exogeneous driving term is a stochatic process.

27A differential equation comprising differential and algebraic terms, given in implicit form. a. DDEb. DAEc. SDEd. SEDb. DAEA differential-algebraic equation (DAE) is an equation involving an unknown function and its derivatives.

28Find the general solution for the differential equation

dy + 7x dx = 0.

a. 2y = -7x + Cb. 7x + 3y + Cc. 2y + 7x + Cd. 3y - 7y + Ca. 2y = -7x + c We simply need to subtract 7x dx from both sides, then insert integral signs and integrate: dy = -7xdx dy = -7xdx ; therefore, y = -7(x/2) + c ; and 2y = -7x + c ; answer.

29What are the other methods in solving linear DE?a. Newton's methodb. Range - Kutta methodsc. Bernoulli's methodd. Euler's methodb. Range - Kutta methodsTwo that are frequently used are Range - Kutta methods.These method is similar to euller's method except that it used 2nd order (parabola).

30Obtain the general solution for the given equation: (1-x)dx = ydya. 2x + y + Cb. y + 2x + Cc. x - x/2 = y/3 + C d. 2x + Cc. x - x/2 = y/3 + C simply integrate both sides, (1-x)dx = ydy ;so x - x/2 = y/3 + C ;Ans.

31What is the solution in the equation sin x sin y dx + cos x cos y dy = 0?a. sin2y + 3x = Cb. siny = Ccosxc. cosy = ysinxd. cosy + 1 = 1b. siny = Ccosxsolution: divide all terms by sin y cos x sin x dx/cos x + cos y dy/sin y = 0 ; integrate: sin x dx/cos x+cos y dy/sin y = In C -In(cos x) + In(sin y) = In C thus: sin y = C cos x ;Answer.

32Solve xydx + edy = 0a. e + y = C b. y = 2x+Cc. 2xy + x = Cd. 2x + 1 = Cya. e + y = C rearrange: xdx/e + dy/y = 0, exdx + ydy = 0 ; integrate: -e(-2xdx) + ydy = 0 e + y/-2 = C/-2 ; thus: e + y = C Answer

33Find the complete solution: 2ydx = 3xdya. 2x + y = Cb. y + 2x = 2Cc. x = Cy d. 2y = x + Cc. x = Cy rearrange: 2dx/x = 3dy/y ;integrate: 2dx/x = 3dy/y, 2 Inx = 3 Iny + In C Inx = Iny + In C In = In(Cy) ; thus: x = Cy Ans

34Solve y' = xy.a. 7x + 3y + Cb. y(x + C)+2 = 0 c. 3y - 7y + Cd. y + 2x = 2Cb. y(x + C)+2 = 0 dy/dx = xy ;thus dy/yxdx, ydy = xdx, integrate: ydy = xdx y/-1 = x/2 + C/2 y(x + C) = -2 ; thus y(x + C)+2 = 0 Answer.

35Find the general solution of dV/dP = -V/P.a. V = PCb. P = CVc. PV = C d. P + 1 = CVc. PV = C thus: dV/V = -dP/P, integrate: dV/V = -dP/P, InV = -InP + In C, InV + InP = In C, In(PV) = In C therefore: PV = C ;Answer.

36Differential equations that involve two or more independent variables a. Partial differential equations.b. Ordinary differential equations.c. Delay differential equations.d. Stochastic differential equations.a. partial differential equations. Differential equations that involve two or more independent variables are called partial differential equations.

37Solve r in the equation dr = b(cosdr + rsind)xa. r = C + cosb. r = sin + Cc. r = C(cos + sin)d. r = C(1 - cos) d. r = C(1 - cos) dr - b cos dr = br sin d , (1 - b cos)dr = r(b sin d ), dr/r = b sin d /1-b cos, dr/r = b sin d /1 - b cos, In r = In(1 - b cos) + In C, In r = In C(1 - b cos) thus: r = C(1 - cos) ;Ans.

38Fid the general solution of the equation xydx - (x + 2)dy = 0a. 2xy = 3xy + Cb. e = Cy(x + 2) c. y = x + cd. None of the above.b. e = Cy(x + 2) xydx = (x + y)dy, xdx/(x + 2) = dy/y, x + 2/x = 1 r.-2, thus: (1 - 2/x + 2)dx = dy/y, (1 - 2/x + 2)dx = dy/y, and so: x - 2In(x + 2) = In y + In C, x = 2In(x + 2) + In Cy(x + 2) thus: Cy(x + 2) = e or e = Cy(x + 2) Ans.

39Solve for the equation x cosydx + tan y dy = 0.a. 3y - 7y + Cb. x + tany = Cc. 2xy = 3xy + Cd. 2y = x + Cb. x + tany = Cxdx + tan y dy/cosy = 0, xdx + tan y secy dy = 0, xdx + tan y sec dy = C/2, x/2 + tany/2 = C/2 ; thus: x + tany = C Answer.

40Find the complete solution of tany dy = sinxdxa. 2y + 7x + Cb. cosx - 3cos = 3(tan y - y + C) c. y(x + C)+2 = 0 d. sinx + 2cos = 3x + Cb. cosx - 3cos = 3(tan y - y + C) (secy - 1)dy = (1 - cosx)sin xdx ; thus: (sec -1)dy = (1 - cosx )sin xdx, tan y - y + C = -cosx + cosx/3 ;and so: cosx - 3cos = 3(tan y - y + C) ;Ans.

41Solve for y' = cosxcos y. a. siny = Ccosxb. y In[C (1-x)] = 1 c. y(x + C)+2 = 0 d. 4In / secy + tany / = 2x +sin2x + Ca. 4In / secy + tany / = 2x +sin2x + Csolution: dy/dx = cosxcos y, dy/cos y = cosxdx, secy dy = 1 + cos/2, thus, sec y dy = 1 + cosx dx/2, In/secy + tany/ = x/2 + sinx + C/4 and so: 4In / secy + tany / = 2x +sin2x + C Answer.

42Obtain the general solution of the equation: (1-x)y' = ya. x+x + Cb. y In[C (1-x)] = 1 c. 2xy = 3xy + Cd. 2y = x + Cb. y In[C (1-x)] = 1 solution: (1-x)dy/dx = y rearranging: dy/y = dx/1-x or ydy = dx/1-x integrate:ydy = dx/1-x 1/y = In(1-x) - InC y In[C (1-x)] thus: y In[C (1-x)] = 1 Answer.

43Who discovered the Integrating factor in 1734?a. Bernoullib. Newtonc. Eulerd. Leibnizc. Euler Integrating factor were discovered by Euler in 1734 and (independently of him) by Fontaine & Clairant though some attribute them Leibniz

44Is a differential equation in which the unknown function is a function of a single independent variablea. Stochasticb. Delay DEc. Partial DEd. Ordinary DEd. Ordinary DEOrdinary DE assumes a continous population variable & does work well when there is a large population of people. Much information canbe obtained.

45Solve (x-4)^4 x(y-3)y'a. (-1/x)+2/x = (-1/y) + 1/y + Cb. x+x + Cc. 3x + y + Cd. None of the abovea. (-1/x)+2/x = (-1/y) + 1/y + CTo separate variables, divide by xy^4 so that, (x-4)dx/x = (y-3)/y^4. this simplifies to (x -4x)dx = (y^-2-3y^-4)dy. Integrating gives: (-1/x)+2/x = (-1/y) + 1/y + C

46Who explained the method of Separating the Variables in Differential Equationa. Isaac Newtonb. John Bernoullic. Charles Darwind. John Lapusb. John BernoulliJhon Bernoulli explained this method in 1694-1697 and he showed he showed how to reduced the homogenous DE of the first order to one w/c the variable were separable.

47This is also referred to as the Cauchy's equation.a. Quadratic Equationb. Bernoulli's Equaionc. Euler's Equationd. Geometryc. Euler's equationAmong the linear equations with the variable coefficient the Euler's equation or Cauchy's equation has been shown to be the more useful one in the engineering applications .

48Find the solution of y(2xy + 1)dx- xdy = 0.a. 2xb. 3x +2y+ Cc. 2y +3Cxd. xy + x = Cyd. xy + x = CySolution: Expand the gven equation: 2xydx + ydx =0 ;divide each term by y, 2xdx + (ydx- xdy)/y = 0 but, (ydx-xdy)/y = d(x/y) so, 2xdx + d(x/y) = 0 ;integration gives, x + x/y = C or xy + x = Cy.

49Find the general solution in the differential equation dy/dx = xy.a. 2lny = x + Cb. x + lnyc. 2xd. lny + 2xa. 2lny = x + CSolution : Separate the variables and then find the antiderivative on each side: dy/y = xdx and lny = x/2 so 2lny = x + C Answer.

50The plot of any solution of differential equation is called what? a. midpointb. integral curvec. -1d. none of the aboveb. Integral curveIt is called the integral curve which is the plot of any solution of differential equation.

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