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SP07Finals-3

# SP07Finals-3 - M427K Final May 7 2007 Linear Equations...

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M427K - Final May 7, 2007 Linear Equations Method of Integrating Factors y prime + p ( t ) y = q ( t ) μ ( t ) = e integraltext p ( t ) dt ( ) prime = μ ( t ) q ( t ) y ( t ) = integraltext μ ( t ) q ( t ) dt + C μ ( t ) Separable Equations When the ODE can be framed in the following form: y prime = f ( t ) g ( y ) dy g ( y ) = f ( t ) dt integraldisplay dy g ( y ) = integraldisplay f ( t ) dt Check if the funstion you are dividing with = 0 or not and check to see if such a function satisfies the original equation. Homogeneous Equations When the ODE can be put in the following form y prime = f ( y/x ) v = y/x y = vx y prime = xv prime + v xv prime + v = f ( v ) xv prime = f ( v ) - v integraldisplay dv f ( v ) - v = integraldisplay dx x Check if solution to f ( v ) = v satifsies the original ODE. 1

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Non-Linear Equations Bernoulli Equations First Order ODEs M ( x, y ) dx + N ( x, y ) dy = 0. Need to find φ ( x, y ) such that φ x ( x, y ) = M ( x, y ) and φ y ( x, y ) = N ( x, y ). Then φ ( x, y ) = C gives the implicit for, of the solution. Exact Equations If M y = N x , the ODE is EXACT. Then φ ( x, y ) = integraltext M ( x, y ) dx + h ( y ) and φ y ( x, y ) = N ( x, y ) = ( integraltext M ( x, y ) dx ) y + h prime ( y ). Then solve for h ( y ) to get the integraltext M ( x, y ) dx + h ( y ) = C . OR If M y = N x , the ODE is EXACT. Then φ ( x, y ) = integraltext N ( x, y ) dy + g ( x ) and φ x ( x, y ) = M ( x, y ) = ( integraltext N ( x, y ) dy ) x + g ( x ). Then solve for g ( x ) to get the integraltext N ( x, y ) dy + g ( x ) = C . OR If M y = N x , the ODE is EXACT. Then φ ( x, y ) = integraltext M ( x, y ) dx + h ( y ) and φ ( x, y ) = integraltext N ( x, y ) dy + g ( x ). Then solve for h ( y ) and g ( x ) to get the integraltext M ( x, y ) dx + h ( y ) = C OR integraltext N ( x, y ) dy + g ( x ) = C . Either of these expressions work. Non-Exact Equations - Integrating Factors If M y negationslash = N x , the ODE is NOT EXACT. Then need to find an integrating factor, which when multiplied to the ODE turns the ODE EXACT. If M y - N x N is a function of x , then the integrating factor is μ ( x ) = exp ( integraltext M y - N x N dx ) OR If M y - N x - M is a function of y , then the integrating factor is λ ( y ) = exp ( integraltext M y - N x - M dy ) Multiply the original equation with the integrating factor thus turning it into an EXACT ODE and then solve it using the known method of solving EXACT ODEs. Second Order Linear ODEs Wronskian and Linear Independence: If y 1 , y 2 are two solutions to the 2nd order ODE, then W ( y 1 , y 2 ) = y 1 y prime 2 - y prime 1 y 2 . If the given ODE is of the form y primeprime + p ( t ) y prime + q ( t ) y = g ( t ), then the Wronskian is given by W ( y 1 , y 2 ) = Cexp ( - integraltext p ( t ) dt ) Homogeneous Equations w/ Constant Coefficients 2
Characteristic Equation in r with real distinct roots r 1 , r 2 General Solution y = C 1 e r 1 t + C 2 e r 2 t Characteristic Equation in r with repeated roots r - Reduction of Order General Solution y = C 1 e rt + C 2 te rt Characteristic Equation in r with complex roots r = λ ± μ General Solution y = e λt ( C 1 cos ( μt ) + C 2 sin ( μt )) Reduction of Order If one of the solutions is known y 1 then. let y 2 = vy 1 . Make this substitution into the original ODE to get a separable ODE in v . Solve this ODE for v to get the other solution y 2 = vy 1 .

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SP07Finals-3 - M427K Final May 7 2007 Linear Equations...

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