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Solutions of Differential Equations
First Method of Solution Variables Separable
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_ dy _ apply arbitrary constant arbitrary function change of variable Chap characteristic curve characteristic strip coefficients complete equation conical element convergence coordinates and slope determine difference dx _ dy dx dy dy dx dy/dx elimination envelope equa exact exact differential example Example.—Solve Exercises.—1 existence theorem expanded family of curves formula geometric given Hence homogeneous identically independent variable indicial equation integral curve integral surface integrating factor interval Legendre polynomials lineal element linear equation linearly dependent Lipschitz condition one-parameter family origin orthogonal trajectories partial differential equation particular solution planar element plane preceding problem Prove reduced equation replaced roots second member second order Show Simpson's rule singular singular solution Solve substituting subtangent tangent theorem Theorem.—The tion vanishes velocity whence Wronskian x-axis yi(x yn(x zero