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sol1 - CAAM 336 DIFFERENTIAL EQUATIONS Problem Set 1...

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CAAM 336 · DIFFERENTIAL EQUATIONS Problem Set 1 · Solutions Posted Wednesday 25 August 2010. Due Wednesday 1 September 2010, 5pm. 1. [18 points: 3 points per part] For each of the following equations, specify whether each is (a) an ODE or a PDE; (b) determine its order; (c) specify whether it is linear or nonlinear. For those that are linear, specify whether they are (d) homogeneous or inhomogeneous, and (e) whether they have constant or variable coefficients. (1.1) dv dx + 2 x v = 0 (1.2) ∂v ∂t - 3 ∂v ∂x = x - t (1.3) ∂u ∂t - ∂x 2 u ∂u ∂x = 0 (1.4) ∂u ∂t + u ∂u ∂x + 3 u ∂x 3 = 0 (1.5) d 2 y dx 2 - μ (1 - y 2 ) dy dx + y = 0 (1.6) d 2 dx 2 ρ ( x ) d 2 u dx 2 = sin( x ) Solution. (1.1) ODE, first order, linear, homogeneous, variable coefficient The 2 /x factor in front of the v is the variable coefficient. (1.2) PDE, first order, linear, inhomogeneous, constant coefficient The x - t term on the right, which does not involve v , makes the equation inhomogeneous. (1.3) PDE, second order, nonlinear Using the product rule for partial derivatives, we can write this equation in the equivalent form ∂u ∂t - 2 ∂u ∂x 2 - 2 u 2 u ∂x 2 = 0 . The second and third terms on the left hand side make this equation nonlinear. (1.4) PDE, third order, nonlinear The u ( ∂u/∂x ) term makes this equation nonlinear. This a version of the famous Korteweg-de Vries (KdV) equation that describes shallow water waves. (1.5) ODE, second order, nonlinear The (1 - y 2 )( dy/dt ) term makes this ODE nonlinear. (1.6) ODE, fourth order, linear, inhomogeneous, variable coefficient Using the product rule for partial derivatives, we can write this equation in the equivalent form d 2 ρ dx 2 d 2 u dx 2 + 2 dx d 3 u dx 3 + ρ ( x ) d 4 u dx 4 = sin( x ) , hence we can see that it is fourth order. This equation, attributed to Euler, describes the deflection of a one-dimensional beam with a static load of sin( x ); ρ ( x ) describes the elasticity of the material that constitutes the beam. 2. [18 points: 6 points per part] Determine whether each of the following functions is a solution of the corresponding differential equation from question 1.
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(a) Is v ( x ) = 1 /x 2 a solution of (1.1) ? (b) Is v ( x, t ) = t ( t + x ) a solution of (1.2) ? (c) Is u ( x, t ) = xe t a solution of (1.3) ? Solution. (a) v ( x ) = 1 /x 2 is a solution of (1.1). To plug v ( x ) = 1 /x 2 into the left-hand side of (1.1), we compute dv/dx = d ( x - 2 ) /dx = - 2 x - 3 .
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  • Fall '09
  • Tompson
  • Boundary conditions, Dirichlet boundary condition, Neumann boundary condition, Dirichlet boundary conditions

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