A First Course in Differential Equations with Applications

a²u Answers to odd-numbered auxiliary equation boundary conditions boundary-value problem c₁ c₁e c₁x c₂ capacitor Cauchy-Euler equation coefficients constant convergence cosh curves defined determine dx dt dx dy dx² dy dx eigenvalues equation of motion equilibrium position Euler EXAMPLE Exercises family of solutions farad formula Fourier series function fundamental set given differential equation graph Hence initial conditions initial-value problem integral interval k₁ k₂ Laplace transform linear equation linearly independent m₁ m₂ mass matrix method Miscellaneous Problems nonhomogeneous obtain odd-numbered problems begin orthogonal trajectories particular solution polynomial r₁ r₂ regular singular point roots second-order Section separation of variables series circuit simple harmonic motion sinh solve the given spring ẞx steady-state superposition principle temperature Theorem variables vector weight Wronskian X₁ x²y y₁ y₁(x zero ηπ ди дх