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FIRSTORDER DIFFERENTIAL EQUATIONS
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applied approximation assume boundary conditions Chapter characteristic equation coefficients compute constant converges uniformly corresponding defined derivative determined direction field diverges dt dt dx/dt dy/dt eigenvalues eigenvectors entries equilibrium points equivalent Euler method EXAMPLE exponential matrix first-order system Fourier series given gives Green's function Hence homogeneous equation homogeneous solution independent solutions infinite series initial conditions initial-value problem integral interval inverse Laplace equation Laplace transform linear combination linear equation linearized system linearly independent modified Euler method multiply n-by-n matrix nonlinear nonzero odd function oscillation partial sums particular solution polynomial power series Proof real numbers result second-order equation Section sequence series converges Show shown in Figure sinh Sketch the graph solution formula solution satisfying square matrix Suppose Taylor expansion Theorem trajectory trigonometric unique solution values variable velocity Verify write written yp(x zero