Solving for the Non-homogeneous EquationDate: 10/21/2001 at 19:07:05 From: S. Parker Subject: Non-homogeneous DE I have the equation: x^2(d^2y/dx^2) - 3x(dy/dx) = 3x^3 I attempted to solve and have the general solution for the homogeneous equation. But I am stuck after that - how do I continue solving for the non-homogeneous equation? Date: 10/24/2001 at 19:48:43 From: Doctor Anthony Subject: Re: Non-homogeneous DE This is an example of Euler's linear equation. Put x = e^t Then x(dy/dx) = dy/dt x^2(d^2y/dx^2) = d^2(y)/dt^2 - dy/dt Substitute into the differential equation d^2(y)/dt^2 - dy/dt - 3(dy/dt) = 3e^(3t) d^2(y)/dt^2 - 4(dy/dt) = 3e^(3t) If dy/dt = p d^2/dt^2 = dp/dt we get dp/dt - 4p = 3e^(3t) (D-4)p = 3.e^(3t) The CF is p = A.e^(4t) The PI is p = 3.e^(3t)/(D-4) = 3.e^(3t)/(3-4) = -3.e^(3t) So the general solution is p = A.e^(4t) - 3.e^(3t) dy/dt = A.e^(4t) - 3.e^(3t) y = (A/4)e^(4t) - (1/9)e^(3t) + E y = C.x^4 - (1/9)x^3 + E - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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