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DE:010117Resolve the startup layer hidden by a quasi-steady sensor model

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Singular Perturbation Initial LayerPointwise Versus Uniform Limit

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Modeling and first-order differential equationsDifferential Equations for Engineers · Mixed review

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DE:010118Separate transport lag from a weak startup layer under ramp input

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Slowly Varying Input AsymptoticsWell Prepared Initial Data

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Modeling and first-order differential equationsDifferential Equations for Engineers · Mixed review

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DE:010119Repair a discontinuous outer approximation at a ramp-rate switch

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Piecewise Singular PerturbationInternal Layer State Matching

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Modeling and first-order differential equationsDifferential Equations for Engineers · Mixed review

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DE:010317Reduce a coupled fast–slow subsystem without losing its startup state

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Fast Slow System ReductionTime Scale Eigenvalue Crosscheck

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Linear systems, eigenstructure, and matrix exponentialsDifferential Equations for Engineers · Mixed review

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Models First OrderDifferential Equations For Engineers Problem Solving

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Modeling and first-order differential equationsDifferential Equations for Engineers · Calculation

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DE:9901011The displacement y(t) (in centimeters) of an unforced mass–spring–damper is governed by Classify the motion as over…

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DE:9901012A particle’s position x(t) (in meters) satisfies the unforced equation Determine whether the damping is overdamped,…

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DE:9901013Solve the initial-value problem Express the unique solution in real-valued form (no complex constants).

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DE:9901014The second-order equation does not contain the unknown w.

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DE:9901015The autonomous equation does not contain the independent variable t.

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DE:9901016Solve the Cauchy–Euler initial-value problem for t>0.

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DE:9901017An undamped oscillator is forced slightly off its natural frequency: Find the unique solution \theta(t).

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DE:9901018Using undetermined coefficients, find the general solution of the resonant equation Then determine the unique solut…

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DE:9901019The underdamped initial-value problem has a unique solution of the phase–amplitude form with R>0 and \delta\in[0,2\…

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DE:9901020Solve the nonhomogeneous Cauchy–Euler equation by first finding the general solution of the associated homogeneous…

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DE:9901021Let y(t) satisfy y'' − 5y' + 6y = 4t² − 7t + 1, with y(0) = 3 and y'(0) = −2.

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DE:9901022The displacement x(t) of a mass–spring system obeys x'' + 9x = 6 cos(3t), with x(0) = 1 and x'(0) = 0.

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DE:9901023Find the general solution of y'' − 4y' + 13y = e^{2t} sin(3t).

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DE:9901024Solve the initial-value problem y'' + y = tan t on the interval 0 < t < π/2, subject to y(π/4) = 2 and y'(π/4) = −1.

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DE:9901025Let u(x) satisfy u'' − 2u' + u = e^x / x² for x > 0.

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DE:9901026A forcing function is piecewise: f(t) = 8 for 0 ≤ t < 2, and f(t) = 0 for t ≥ 2.

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DE:9901027The unknown w(τ) satisfies w'' − 3w' = 6τ + 4, with w(0) = −1 and w'(0) = 5.

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DE:9901028Define the piecewise forcing g(t) by g(t) = t for 0 ≤ t < 1, g(t) = 2 − t for 1 ≤ t < 2, and g(t) = 0 for t ≥ 2.

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DE:9901029For the constant-coefficient operator L[y] = y'' − y, construct the causal Green's function G(t, s) (that is, G(t,…

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DE:9901030Consider the homogeneous linear system x' = -7x + 2y, y' = 3x - 8y, where x = x(t) and y = y(t).

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DE:9901031Let u = u(t) and v = v(t) satisfy u' = 5u + 6v, v' = 4u - 3v.

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DE:9901032The planar system p' = 4q, q' = -9p has an isolated equilibrium at the origin in the pq-plane.

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DE:9901033Let x1 = x1(t) and x2 = x2(t) satisfy x1' = -2 x1 - 5 x2, x2' = x1 - 4 x2.

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DE:9901034Consider r' = 3r + 4s, s' = -4r + 3s, where r = r(t) and s = s(t).

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DE:9901035The coefficient matrix of the system w' = 5w + 4z, z' = -1w + 1z has a repeated eigenvalue.

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DE:9901036Let X(t) = (x(t), y(t))^T satisfy X' = A X, where A = [ 0 -3 ] [ 1 4 ].

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DE:9901037Solve the nonhomogeneous system x' = -x + 2y + 4 e^{-t}, y' = 3x - 2y + 2 e^{2t} by variation of parameters.

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DE:9901038The linear system a' = -6a, b' = -6b has a repeated eigenvalue that is not defective.

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DE:9901039Two vector functions X^{(1)}(t) = e^{-t} ( 1, 1 )^T, X^{(2)}(t) = e^{2t} ( 1, -2 )^T are proposed as solutions of a…

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DE:9901040Let F(s)=\mathcal{L}\{f(t)\}(s)=\dfrac{3s+11}{(s+2)^2(s^2+9)}.

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DE:9901041Let g(t)=(t-4)^2 e^{-3(t-4)}\,u(t-4), where u denotes the unit step function.

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DE:9901042Let y(t) satisfy the initial-value problem Compute Y(s)=\mathcal{L}\{y(t)\}(s), then recover y(t) by completing the…

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DE:9901043The function h(t) is defined piecewise by Write h(t) in terms of the unit step function, compute H(s)=\mathcal{L}\{…

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DE:9901044Solve the initial-value problem where \delta is the Dirac delta.

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DE:9901045Let f(t)=3t\,e^{-t} and g(t)=\sin(2t).

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DE:9901046A forcing function r(t) equals 8 on the delayed window 3\le t< 7 and equals 0 elsewhere.

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Problem library

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Topic choices for Differential Equations for Engineers are shortened. Search still covers every problem.

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DE:010117Resolve the startup layer hidden by a quasi-steady sensor model

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Singular Perturbation Initial LayerPointwise Versus Uniform Limit

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Modeling and first-order differential equationsDifferential Equations for Engineers · Mixed review

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DE:010118Separate transport lag from a weak startup layer under ramp input

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Slowly Varying Input AsymptoticsWell Prepared Initial Data

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Modeling and first-order differential equationsDifferential Equations for Engineers · Mixed review

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DE:010119Repair a discontinuous outer approximation at a ramp-rate switch

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Piecewise Singular PerturbationInternal Layer State Matching

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Modeling and first-order differential equationsDifferential Equations for Engineers · Mixed review

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DE:010317Reduce a coupled fast–slow subsystem without losing its startup state

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Fast Slow System ReductionTime Scale Eigenvalue Crosscheck

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Linear systems, eigenstructure, and matrix exponentialsDifferential Equations for Engineers · Mixed review

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Models First OrderDifferential Equations For Engineers Problem Solving

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Modeling and first-order differential equationsDifferential Equations for Engineers · Calculation

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Models First OrderDifferential Equations For Engineers Problem Solving

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DE:9901011The displacement y(t) (in centimeters) of an unforced mass–spring–damper is governed by Classify the motion as over…

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Models First OrderDifferential Equations For Engineers Problem Solving

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DE:9901012A particle’s position x(t) (in meters) satisfies the unforced equation Determine whether the damping is overdamped,…

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DE:9901013Solve the initial-value problem Express the unique solution in real-valued form (no complex constants).

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DE:9901014The second-order equation does not contain the unknown w.

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DE:9901015The autonomous equation does not contain the independent variable t.

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DE:9901016Solve the Cauchy–Euler initial-value problem for t>0.

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DE:9901017An undamped oscillator is forced slightly off its natural frequency: Find the unique solution \theta(t).

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DE:9901018Using undetermined coefficients, find the general solution of the resonant equation Then determine the unique solut…

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DE:9901019The underdamped initial-value problem has a unique solution of the phase–amplitude form with R>0 and \delta\in[0,2\…

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DE:9901020Solve the nonhomogeneous Cauchy–Euler equation by first finding the general solution of the associated homogeneous…

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DE:9901021Let y(t) satisfy y'' − 5y' + 6y = 4t² − 7t + 1, with y(0) = 3 and y'(0) = −2.

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DE:9901022The displacement x(t) of a mass–spring system obeys x'' + 9x = 6 cos(3t), with x(0) = 1 and x'(0) = 0.

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DE:9901023Find the general solution of y'' − 4y' + 13y = e^{2t} sin(3t).

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DE:9901024Solve the initial-value problem y'' + y = tan t on the interval 0 < t < π/2, subject to y(π/4) = 2 and y'(π/4) = −1.

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DE:9901025Let u(x) satisfy u'' − 2u' + u = e^x / x² for x > 0.

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DE:9901026A forcing function is piecewise: f(t) = 8 for 0 ≤ t < 2, and f(t) = 0 for t ≥ 2.

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DE:9901027The unknown w(τ) satisfies w'' − 3w' = 6τ + 4, with w(0) = −1 and w'(0) = 5.

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DE:9901028Define the piecewise forcing g(t) by g(t) = t for 0 ≤ t < 1, g(t) = 2 − t for 1 ≤ t < 2, and g(t) = 0 for t ≥ 2.

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DE:9901029For the constant-coefficient operator L[y] = y'' − y, construct the causal Green's function G(t, s) (that is, G(t,…

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DE:9901030Consider the homogeneous linear system x' = -7x + 2y, y' = 3x - 8y, where x = x(t) and y = y(t).

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DE:9901031Let u = u(t) and v = v(t) satisfy u' = 5u + 6v, v' = 4u - 3v.

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DE:9901032The planar system p' = 4q, q' = -9p has an isolated equilibrium at the origin in the pq-plane.

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DE:9901033Let x1 = x1(t) and x2 = x2(t) satisfy x1' = -2 x1 - 5 x2, x2' = x1 - 4 x2.

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DE:9901034Consider r' = 3r + 4s, s' = -4r + 3s, where r = r(t) and s = s(t).

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DE:9901035The coefficient matrix of the system w' = 5w + 4z, z' = -1w + 1z has a repeated eigenvalue.

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DE:9901036Let X(t) = (x(t), y(t))^T satisfy X' = A X, where A = [ 0 -3 ] [ 1 4 ].

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DE:9901037Solve the nonhomogeneous system x' = -x + 2y + 4 e^{-t}, y' = 3x - 2y + 2 e^{2t} by variation of parameters.

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DE:9901038The linear system a' = -6a, b' = -6b has a repeated eigenvalue that is not defective.

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DE:9901039Two vector functions X^{(1)}(t) = e^{-t} ( 1, 1 )^T, X^{(2)}(t) = e^{2t} ( 1, -2 )^T are proposed as solutions of a…

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DE:9901040Let F(s)=\mathcal{L}\{f(t)\}(s)=\dfrac{3s+11}{(s+2)^2(s^2+9)}.

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DE:9901041Let g(t)=(t-4)^2 e^{-3(t-4)}\,u(t-4), where u denotes the unit step function.

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DE:9901042Let y(t) satisfy the initial-value problem Compute Y(s)=\mathcal{L}\{y(t)\}(s), then recover y(t) by completing the…

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DE:9901043The function h(t) is defined piecewise by Write h(t) in terms of the unit step function, compute H(s)=\mathcal{L}\{…

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DE:9901044Solve the initial-value problem where \delta is the Dirac delta.

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DE:9901045Let f(t)=3t\,e^{-t} and g(t)=\sin(2t).

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DE:9901046A forcing function r(t) equals 8 on the delayed window 3\le t< 7 and equals 0 elsewhere.

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Problem results

DE:9901001A 400-liter tank initially contains 80 liters of brine with 12 kg of dissolved salt.

DE:9901002A copper rod is heated so that its temperature is 190°C and is then placed in a room whose air temperature remains…

DE:9901003A cylindrical tank of radius 1.5 m stands on end and is initially filled with water to a depth of 2.4 m.

DE:9901004A sample of the radioisotope iodine-131 has an initial mass of 48.0 mg.

DE:9901005In a closed woodland, the deer population P(t) (in hundreds) obeys the logistic equation P' = 0.32 P (1 − P/18), wi…

DE:9901006A series RC circuit has resistance R = 2.4 × 10³ Ω and capacitance C = 5.0 × 10⁻⁶ F.

DE:9901007Consider the family of curves y = k x² e^{−x} for k ≠ 0.

DE:9901008Consider the differential equation (3x² y + 8 x y² − 4) dx + (x³ + 8 x² y + 6 y) dy = 0.

DE:9901009The linear equation (x² + 1) dy/dx + 4 x y = 6 x is not already in standard form.

DE:9901010The first-order equation (2 x y + 3 y²) dx − (x² + 2 x y) dy = 0 is homogeneous of degree 2.

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Problem results

DE:9901001A 400-liter tank initially contains 80 liters of brine with 12 kg of dissolved salt.

DE:9901002A copper rod is heated so that its temperature is 190°C and is then placed in a room whose air temperature remains…

DE:9901003A cylindrical tank of radius 1.5 m stands on end and is initially filled with water to a depth of 2.4 m.

DE:9901004A sample of the radioisotope iodine-131 has an initial mass of 48.0 mg.

DE:9901005In a closed woodland, the deer population P(t) (in hundreds) obeys the logistic equation P' = 0.32 P (1 − P/18), wi…

DE:9901006A series RC circuit has resistance R = 2.4 × 10³ Ω and capacitance C = 5.0 × 10⁻⁶ F.

DE:9901007Consider the family of curves y = k x² e^{−x} for k ≠ 0.

DE:9901008Consider the differential equation (3x² y + 8 x y² − 4) dx + (x³ + 8 x² y + 6 y) dy = 0.

DE:9901009The linear equation (x² + 1) dy/dx + 4 x y = 6 x is not already in standard form.

DE:9901010The first-order equation (2 x y + 3 y²) dx − (x² + 2 x y) dy = 0 is homogeneous of degree 2.