M285_2.3_Linear+equations.pdf - Math 285 2.3 Linear Equations Definition A first-order differential equation of the form dy a1 x a0 x y g x dx is said

M285_2.3_Linear+equations.pdf - Math 285 2.3 Linear...

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Math 285: 2.3 Linear Equations Definition: A first-order differential equation of the form 1 0 ( ) ( ) ( ) dy a x a x y g x dx ± is said to be a linear equation in one variable y. Standard Form: ( ) ( ). dy P x y f x dx ± Integrating factor: ( ) P x dx e ³ A transient term of the ODE is a term such that upon taking the limit as x ĺLQILQLW\ WKH WHUP WHQGV WR ±² $ QRQ transient term that tends to a constant is called a steady state term. Solving a Linear First-Order Equation (i) Remember to put a linear equation into standard form. (ii) From standard form of the equation identify P(x) and then find integrating factor ( ) P x dx e ³ . No constant need be used in evaluating the indefinite integral ( ) . P x dx ³ (iii) Multilpy the both sides of the standard form equation by the integrating factor. The left-hand side of the resulting equation is automatically the derivative of the product of the integrating factor ( ) P x dx e ³ and y: ( ) ( ) ( ). P x dx P x dx d e y e f x dx ª º ³ ³ « » ¬ ¼ (iv) Integrate both sides of the last equation and solve for y. Example 1: Solve the differential equation 2 2 3 6 . dy x y x dx ± - l im PCX ) O xx Standard Form ( easy , cos X dat - - e Mz} cost PCXZ ) - - 3 × 2 fcx Y ( O ) "
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