Differential Equations |
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a₁ arbitrary constants assume b₁ beam c₁ c₁e c₁x c₂ Chapter characteristic equation circuit Clairaut's equation coefficients convergence cosh d2x dt2 d2x dx d³y deflection curve dependent variable derivatives determine differential equation dt2 dt dx/dt dy dx dy/dx envelope equa equation becomes equation of motion Euler's formula exact differential EXAMPLE EXERCISE existence theorem family of curves force ft./sec function given gives Hence homogeneous equation initial conditions integral curves integrating factor linear equation mass method obtain original equation orthogonal trajectories particular integral particular solution poundals problem recursion formula resistance satisfy Show singular solutions sinh slope Solve the equation spring student substitution temperature theorem tion transformation v₁ values write x-axis x)dy x²y x³y y)dx y)dy y₁ zero