 1.1.1: In each of 1 through 6, determine whether y = (x) is a solution of ...
 1.1.35: In each of 1 through 5, find the general solution.y 3x y = 2x 2
 1.1.48: In each of 1 through 5, test the differential equation for exactnes...
 1.1.64: In each of 1 through 14, find the general solution. These problems ...
 1.1.83: A 10pound ballast bag is dropped from a hot air balloon which is a...
 1.1.105: In each of 1 through 4, use Theorem 1.2 to show that the initial va...
 1.1.2: In each of 1 through 6, determine whether y = (x) is a solution of ...
 1.1.36: In each of 1 through 5, find the general solution.y+ y = 12 (ex ex )
 1.1.49: In each of 1 through 5, test the differential equation for exactnes...
 1.1.65: y+ 1 x = 2 x 3 y4/3
 1.1.84: A 48 pound box is given an initial push of 16 feet per second down ...
 1.1.106: y= ln x y; y(3) =
 1.1.3: In each of 1 through 6, determine whether y = (x) is a solution of ...
 1.1.37: In each of 1 through 5, find the general solution.y+ 2y = x
 1.1.50: In each of 1 through 5, test the differential equation for exactnes...
 1.1.66: y+ x y = x y2
 1.1.85: A skydiver and her equipment together weigh 192 pounds. Before the ...
 1.1.107: y= x 2 y2 + 8x/y; y(3) = 1
 1.1.4: In each of 1 through 6, determine whether y = (x) is a solution of ...
 1.1.38: In each of 1 through 5, find the general solution.y+ sec(x)y = cos(x)
 1.1.51: In each of 1 through 5, test the differential equation for exactnes...
 1.1.67: y= x y + y x
 1.1.86: Archimedes principle of buoyancy states that an object submerged in...
 1.1.108: y= cos(ex y ); y(0) = 4
 1.1.5: In each of 1 through 6, determine whether y = (x) is a solution of ...
 1.1.39: In each of 1 through 5, find the general solution.y 2y = 8x 2
 1.1.52: In each of 1 through 5, test the differential equation for exactnes...
 1.1.68: y= y x + y
 1.1.87: Suppose the box in cracks open upon hitting the bottom of the lake,...
 1.1.109: Consider the initial value problem y  = 2y; y(x0) = y0, in which ...
 1.1.6: In each of 1 through 6, determine whether y = (x) is a solution of ...
 1.1.40: In each of 6 through 10, solve the initial value problem.y+ 3y = 5e...
 1.1.53: In each of 6 and 7, determine so that the equation is exact. Obtain...
 1.1.69: y= 1 2x y2 1 x y 4 x
 1.1.88: The acceleration due to gravity inside the earth is proportional to...
 1.1.110: You should find that the iterates computed in part (c) are exactly ...
 1.1.7: In each of 7 through 16, determine if the differential equation is ...
 1.1.41: In each of 6 through 10, solve the initial value problem.y+ 1x2 y =...
 1.1.54: In each of 6 and 7, determine so that the equation is exact. Obtain...
 1.1.70: (x 2y)y= 2x y
 1.1.89: A particle starts from rest at the highest point of a vertical circ...
 1.1.111: y= 4 + y; y(0) = 3
 1.1.8: In each of 7 through 16, determine if the differential equation is ...
 1.1.42: In each of 6 through 10, solve the initial value problem.y y = 2e4x...
 1.1.55: In each of 8 through 11, determine if the differential equation is ...
 1.1.71: x y= x cos(y/x) + y
 1.1.90: Determine the currents in the circuit of Figure 1.13
 1.1.112: y= 2x 2 ; y(1) = 3
 1.1.9: In each of 7 through 16, determine if the differential equation is ...
 1.1.43: In each of 6 through 10, solve the initial value problem.y+ 2x+1 y ...
 1.1.56: In each of 8 through 11, determine if the differential equation is ...
 1.1.72: y+ 1 x y = 1 x 4 y3/4
 1.1.91: In the circuit of Figure 1.14, the capacitor is initially discharge...
 1.1.113: y= cos(x); y( ) = 1
 1.1.10: In each of 7 through 16, determine if the differential equation is ...
 1.1.44: In each of 6 through 10, solve the initial value problem.y+ 5y9x = ...
 1.1.57: In each of 8 through 11, determine if the differential equation is ...
 1.1.73: x 2 y= x 2 + y2
 1.1.92: For the circuit in Figure 1.15, find all currents immediately after...
 1.1.11: In each of 7 through 16, determine if the differential equation is ...
 1.1.45: Find all functions with the property that the y intercept of the ta...
 1.1.58: In each of 8 through 11, determine if the differential equation is ...
 1.1.74: y= 1 x y2 + 2 x y
 1.1.93: In a constant electromotive force RL circuit, we find that the curr...
 1.1.12: In each of 7 through 16, determine if the differential equation is ...
 1.1.46: A 500 gallon tank initially contains 50 gallons of brine solution i...
 1.1.59: In each of 8 through 11, determine if the differential equation is ...
 1.1.75: x 3 y= x 2 y y3
 1.1.94: Recall that the charge q(t) in an RC circuit satisfies the linear d...
 1.1.13: In each of 7 through 16, determine if the differential equation is ...
 1.1.47: Two tanks are connected as in Figure 1.6. Tank 1 initially contains...
 1.1.60: Let be a potential function for M + N y= 0. Show that + c is also a...
 1.1.76: y= ex y2 + y + ex
 1.1.95: In each of 13 through 17, find the family of orthogonal trajectorie...
 1.1.14: In each of 7 through 16, determine if the differential equation is ...
 1.1.61: (a) Show that y x y =0 is not exact on any rectangle in the plane. ...
 1.1.77: y+ 2 x y = 3 x y2
 1.1.96: x + 2y = k
 1.1.15: In each of 7 through 16, determine if the differential equation is ...
 1.1.62: Show that x 2 y + x y = y3/2is not exact. Solve this equation by fi...
 1.1.78: Consider the differential equation y = F ax + by + c dx + py +r in ...
 1.1.97: y = kx 2 + 1
 1.1.16: In each of 7 through 16, determine if the differential equation is ...
 1.1.63: Try the strategy of on the differential equation 2y2 9x y + (3x y 6...
 1.1.79: y= y 3 x + y 1
 1.1.98: x 2 + 2y2 = k
 1.1.17: In each of 17 through 21, solve the initial value problem.x y2 y= y...
 1.1.80: y= 3x y 9 x + y + 1
 1.1.99: y = ekx
 1.1.18: In each of 17 through 21, solve the initial value problem.y= 3x 2(y...
 1.1.81: y= x + 2y + 7 2x + y 9
 1.1.100: A man stands at the junction of two perpendicular roads, and his do...
 1.1.19: In each of 17 through 21, solve the initial value problem.ln(yx )y=...
 1.1.82: y= 2x 5y 9 4x + y + 9
 1.1.101: A bug is located at each corner of a square table of side length a....
 1.1.20: In each of 17 through 21, solve the initial value problem.2yy= exy2...
 1.1.102: A bug steps onto the edge of a disk of radius a that is spinning at...
 1.1.21: In each of 17 through 21, solve the initial value problem.yy= 2x se...
 1.1.103: A 24 foot chain weighing pounds per foot is stretched out on a very...
 1.1.22: An object having a temperature of 90 Fahrenheit is placed in an env...
 1.1.104: Suppose the chain in is placed on a table that is only 4 feet high,...
 1.1.23: A thermometer is carried outside a house whose ambient temperature ...
 1.1.24: A radioactive element has a halflife of ln(2) weeks. If e3 tons ar...
 1.1.25: The halflife of Uranium238 is approximately 4.5(109 ) years. How ...
 1.1.26: Given that 12 grams of a radioactive element decays to 9.1 grams in...
 1.1.27: Evaluate 0 et29/t2 dt. Hint: Let I(x) = 0 et2(x/t)2 dt. Calculate I...
 1.1.28: (Draining a Hot Tub) Consider a cylindrical hot tub with a 5foot r...
 1.1.29: Calculate the time required to empty the hemispherical tank of Exam...
 1.1.30: Determine the time it takes to drain a spherical tank with a radius...
 1.1.31: A tank shaped like a right circular cone, vertex down, is 9 feet hi...
 1.1.32: Determine the rate of change of the depth of water in the tank of (...
 1.1.33: (Logistic Model of Population Growth) In 1837, the Dutch biologist ...
 1.1.34: Continuing 33, a 1920 study by Pearl and Reed (appearing in the Pro...
Solutions for Chapter 1: FirstOrder Differential Equations
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 1: FirstOrder Differential Equations
Get Full SolutionsSince 113 problems in chapter 1: FirstOrder Differential Equations have been answered, more than 26639 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1: FirstOrder Differential Equations includes 113 full stepbystep solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Iterative method.
A sequence of steps intended to approach the desired solution.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.