- Given thatparticular solution. has initial conditionswhen x = 0, find the particular solution. d²y dx² dy 5+4y=x²+e-2x dx
- Let z = cos θ + isin θ. Show thatBy considering (1 + z)n, where n is a positive integer, deduce the sum of the series 1+z = 2cos-0 cos -0 + isin-0
- Simplify cos5 θ + isin5 θ, .giving your answer in an exponential form, where the terms are of the form aekiθ.
- The points A, B, C have position vectors ai, bj, ck respectively, where a, b, c are all positive.The plane counting A, B, C is denoted ∏.i. Find a vector perpendicular to ∏.ii. Find the
- The variable y depends on x and the variables x and t are related by x = 1/t. Show thatThe variables x and y are related by the differential equationShow thatHence find the general solution for y in
- Find the eigenvalues and corresponding eigenvectors of the matrix A, whereHence find a non-singular matrix P and a diagonal matrix D such that A + A2 + A3 = PDP-1. 4 1) A = -6 -1 3 84
- Write down an expression in terms of z and N for the sum of the seriesUse de Moiver’s theorem do deduce that N Σ 2 "z
- Given that x = t3 + 1, y = t2 – t, find d2y/dx2.
- The matrixIs such that B2 = A. Using diagonalisation, or otherwise, find a matrix B that satisfies the condition above. 33 24 A = 48 57
- A particle, of mass 2 kg, is dropped from rest and falls towards the ground. The resistance to motion is modelled by 5vN, where v is the velocity of the particle at any given time. The displacement
- Find the general solution for the differential equation d?r dr +4 + 3r = 2e-0 + e-30. de? de
- A substance, X, is decaying such that the rate of change of the mass, x, is proportional to four times its mass. Another substance, Y, is such that the rate of change of the mass, y, is equal to x -
- Show that cos6 θ can be written in the form a cos 6θ + b cos 4θ + ccos 2θ + d. Hence, find cos 0 d0.
- The variables x and u are related by ux = 2. The variable y is dependent on x.a. Show thatb. Show also thatreduces to 16d2y/du2 – 8 dy/du + y = u and find the general solution in the form y =
- Find the general solution for the differential equation d2y/dx2 + y = x – cosx. Given that the initial conditions are y = 1, dy/dx = 1 when x = 0, find the particular solution.
- Given that d2s/dθ2 + 7 ds/dθ + 6s = 0 has initial conditions s = 1, ds/dθ = -1 when θ = 0, show that s → 0 for large values of θ, and sketch the curve representing the particular solution.
- Given that the differential equation sect dy/dt + ycosec t = sin2 tsect has initial conditions y = 0, t = π/2, find the particular solution.
- Given thatshow thatHence, evaluate 1, = sinh" xdx,
- The variables z and x are related such thatGiven that y = 2z3, find a differential equation relating y and x.Hence, determine the general solution in the form z = f(x). 2 dz + 2z \dx, + z2 d2 - 22 =
- The initial conditions for the differential equationa. Show that the particular integral must be of the form h = αt2e1/2t and find the value of α.b. Hence, determine the particular solution. d?h 4.
- A differential equation is given as d2y/dx2 + 4y = 0.a. Find the complementary function.b. Using the boundary conditions x = 0, y = 1 and x = π/2, dy/dx = -1, show that the particular solution is of
- Find the general solution for the differential equation dy/dx + y tanhx = 4x.
- a. If y = cosh-1 (2x - 3), show thatb. Given that ex+2Y = xy + 1 passes through (0, 0), find the value of d2y/dx2 at that point. dy 1 dx 2(x - 1)(x- 2) II
- The variable y depends on x, and x and t are related by x = e2t.a. Show that 2x dy/dx = dy/dt and find d2y/dx2.b. Find the general solution for 4x2 d2y/dx2 + 8x dy/dx + y = 3x.
- Given that d2x/dt2 + 5 dx/dt + 4x = e-t, find the general solution.
- Find the particular solution for the second order differential equation 9d2r/dt2 – 12 dr/dt + 4r = 0, given that the initial conditions are r = 2 and dr/dt = 3 when t = 0.
- A differential equation is such that the rate of change of y with respect to x is equal to the sum of x and y.a. Write down the differential equation.b. Find the general solution.
- The matrix P has eigenvector e with corresponding eigenvalue 2, and the matrix Q also has eigenvector e with corresponding eigenvalue µ.Find the corresponding eigenvalues and eigenvectors for the
- Given thatShot that y = z/x2 reduces this differential equation toHence, find the general solution z = f(x). 1 d?z x² dx? 3 4 dz Z = 5x – 6 xdx
- State, without evaluating, the correct form for the particular integral in the following cases.a.b.c. d'y dy +3 - 4y = 5e* dx2 dx
- For the differential equation d2x/dt2 determine the particular solution, given that the initial conditions are x = 1 and dx/dt = 4 when t = 0.
- Solve the differential equation when (x – 2)dy/dx – 2y = (x – 2)4 and find the particular solution, given that y = 4 when x – 0.
- a. You are given that y = cosech-1 2x. Show that whenb. Find the exact solutions of 8 coshx - 7 sinh x = 4. d?y x< 0, dx xVx + 1 2
- In each of the following cases, determine the first and second derivatives for y.a. y = 2xw, where w is a function of xb. y = z5, where both y and z are functions of xc. x = 1/t2, where the variable
- Find the general solution of d2y/dx2 + 6dy/dx + 9y = x + e2x, stating limx→∞ y.
- For the second order differential equation a d2y/dx2 + bdy/dx + cy = 0, write the complementary functions for each of the following cases.a. λ1 = 2, λ2 = -5b. λ1 = -2 + 3i, λ2 = -2 -3ic. λ = 4
- The differential equation 2 dx/dt -4x = t has initial conditions x = 1, t = 0. Find the particular solution.
- Show that, with a suitable value of the constant α, the substitution y = xαw reduces the differential equationFind the general solution for y in the case where f(x) = 6 sin 2x + 7 cos 2x. + (3x +
- Solve the differential equation dy/dx = cos(x + y) using the substitution u = x + y, giving your answer as a general solution.
- Given that d2x/dt2 + 2 dx/dt + 5x = et + 2 has initial conditions x = -2, dx/dt = 0 when t = 0, find the particular solution.
- Find the general solution for the differential equation d2y/dx2 + 16dy/dx + 64y = 0.
- Solve the differential equation dy/dx – 7y = e2x to find the general solution.
- Find the value of the constant k such that y = kx2e2x is a particular integral of the differential equationHence find the general solution of (*).Find the particular solution of (*) such that y = 3
- Find the general solution for the differential equation dy/dx = y2 – x2/yx, using the substitution u = x/y.
- The second order differential equation d2r/dt2 – 4r = 3cost has initial conditions r = 1, dy/dt = 1 when t = 0.a. Find the complementary function and the particular integral.b. Hence, using the
- Find the general solution for the differential equation d2y/dx2 – 6dy/dx – 34y = 0.
- Solve the differential equation dy/dx + 2y = x3 e-2x to find the general solution.
- A first order differential equation is given as (3x + 4)dy/dx – 3y = x. Determine the general solution of this differential equation.Given that y = 13/9 when x = - 1, find the value of the constant
- Using the substitution u = e-y, solve the differential equation dy/dx = -(1 – e-y).
- Find the general solution for d²y dy + 5 + 6y = -2x2. dx? dx
- Find the general solution for the differential equation d2y/dx2 – 3dy/dx – 18y = 0.
- Solve the differential equation dy/dx + 2/x y = sinx/x2 find the general solution.
- The polynomial P is given by (z + 2 - 3i)3 = 2 + 2i. Given that ω = z + 2 - 3i, find the roots ω, ω2, ω3 and state, in terms of the z-plane, the value of ω + ω2 + ω3. Sketch the three
- a. Write down the sixth roots of unity in exponential form.b. State the two real roots.c. The polynomial z6 - 1 = 0 can be written as the product of two linear factors and two quadratic factors. Show
- By considering thatfind all solutions for the polynomial 2x5 - 5x4 - 20x3 + 10x2 + 10x = 1. 5 tan 0 - 10 tan' e + tan 0 1 - 10 tan? 0 + 5 tan e tan 50 4
- By consideringAnd then using only the imaginary part of your result, show that: Σ z2n-1 n=1
- Show that (z - eiθ)(z – e-iθ) = z2 - 2zcos θ + 1.
- Find, using complex number methods, the value ofgiving your answer as an exact value. sin'e de,
- Show thatHence, determine the solutions to the equation x4 - 4x3 - 6x2 + 4x = - 1. 4 tan 0 – 4 tan' 0 tan 40 = 1 - 6 tan? 0 + tan e
- Determine the value ofGiving your answer in an exact form. Σ Cos 3 n=1 /3
- Find the roots of Z12 = -1. Give your answers in exponential form.
- Find cos8 θ + sin8 θ in the form acosbθ + ccosdθ + e.
- Find sin θ/cos θ in terms of sin θ.
- By first expanding,Show that: N-1 n=0
- Solve the equation z8 = -8√3 + 8i. Show all these solutions on an Argand diagram.
- Show thatcan be written as cos θ(2 cos θ - sec θ - 2i sin θ). (cos0 + isin 0)6 (cos0 +isine)8
- Show that, when N is even, the real part ofIs equal to ksin2 θ(tanN θ – 1), where k = ± 1. N 2(1 - sec ee")" 1=1
- A polynomial equation is given as z5 - 1 = 0.a. Show that I is one root, then factorise your polynomial into the form (z - 1)f(z) = 0, where f(z) is to be stated.b. Using the four complex solutions,
- Find cos3 θ sin3 θ in terms of sin 6θ and sin 2θ.
- Find tan 3θ in terms of tan θ and, hence, solve the equation 2x3 + 9x2 - 3 = 6x.
- a. Determinein terms of multiple angles of θ.b. State the value of 10 >4 sin (2n – 1)0 2=1
- Find the 4th roots of -16, giving each root in an exponential form.
- Find 4cos5 B in terms of cosines of multiple angles.
- Using de Moivre's theorem, or otherwise, find cos 7θ in terms of cos θ.
- Let ω = cos 1/5 π + is in 1/5 π. Show that ω5 + 1 = 0 and deduce that ω4 – ω3 + ω2 – ω = -1.Show further that ω – ω4 = 2 cos 1/5 π and ω3 – ω2 = 2cos 3/5 π.Hence find the values
- ExpandGiving your answer in trigonometric form. N Σ z2n- 2n-1 N=1
- Find the roots of the equation z4 = 8 + 8√3i, giving your answers in exponential form.
- Find the values of the following expressions giving your answer as an exact value.a.b. (√3 + i)12c. (1/1 + i)6 +isin 12 co- Cos 12,
- Use de Moivre's theorem to express cot 7θ in terms of cot θ.(Use the equation cot 7θ = 0 to show that the roots of the equation x6 - 21x4 + 35x2 - 7 = 0 are cot (1/14 kπ) for k = 1, 3, 5, 9, 11,
- Show thatCan be written as: N 23" z" n=1
- The roots of z3 - 1 = 0 are 1, ω, ω2. Determine the value of ω3 + ω4 + ω5. Otherwise, ω3 = 1, ω4 = ω and ω5 = ω2.
- Write cos 4xcosx in terms of powers of cos x.
- Given that z = eiθ, determine an expression forIn terms of exponentials. N Σ z" n=1
- Find the 6th roots of unity. Give your answers in integer or exponential form.
- Find sin4 θ + 4 cos4 θ in terms of cosines of multiple angles.
- Find the values of the following complex numbers.a. (-√3 – i)14b. (-1 + i)9
- Using the Cayley-Hamilton theorem, find the inverse of the matrix 0 -4 -6 0 -3 A = -1 1 2 5,
- The point P(2, 1) lies on the curve with equation x3 - 2y3 = 3xy.Findi. The value of dy/dx at P,ii. The value d2y/dx2 at P.
- Given that y = x sinh-1 2x, find d2y/dx2.
- Determine the integral dx. 2+x² 2.
- Solve 19sinhx + 16coshx = 8.Write all solutions in exact form unless otherwise stated.
- Prove that cosh2x = cosech2x + 1/coth2x – 1
- Find the exact value of:a. sinh-1 3b. tanh-1 2/3c. cosh-1 5/4
- Solve 5 coshx - cosh 2x = 3. Leave your answer in exact logarithmic form.
- Solve 29 cosh x = 11 sinh x + 27.Write all solutions in exact form unless otherwise stated.
- Prove that cosh3x = 4cosh3x – 3coshx.
- Solve, giving your answers as logarithms.a. 3 cosh2x = -10sinhxb. tanh2x + 5 sechx – 5 = 0c. sin2x = coshx
- Starting from the definitions of sinhx and coshx in terms of exponentials, prove that: a. 1 + 2sinh2x = cosh 2x b. 2 cosh 2x + sinh x = 5
- Solve 17 sinhx + 16 coshx = 8.Write all solutions in exact form unless otherwise stated.
- Prove that coth2x – cothx = ½(tanhx – cothx)