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Unformatted text preview: MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 16 Second Order Differential Equations Linear Equations. We briefly recall some facts about differential equations which we have seen over the past few weeks. Recall that we use the notation dn y y ( n) = n dt to denote the nth derivative. We say that an equation of the form ￿ ￿ F t, y, y (1) , . . . , y (n) = 0 is an nth order differential equation. In particular, we can express the highest order derivative in terms of the lower order derivatives: ￿ ￿ dn y = G t, y, y (1) , . . . , y (n−1) dtn for some function G(t, y1 , y2 , . . . , yn ). When n = 2, we call an equation of the form ￿ ￿ d2 y dy = G t, y, dt2 dt a second order differential equation. Recall that a first order differential equation is said to be a linear equation if it is in the form dy = G(t, y ) where G(t, y ) = g (t) − p(t) y dt for some functions p(t) and g (t). Similarly, we say that a second order differential equation is a linear equation if it is in the form ￿ ￿ d2 y dy dy = G t, y, where G (t, y1 , y2 ) = g (t) − q (t) y − p(t) dt2 dt dt for some functions p(t), q (t), and g (t). Any second order differential equation which cannot be placed in this form is called a nonlinear equation. Equivalently, we say that a second order differential equation is linear if it is in the form d2 y dy + p(t) + q (t) y = g (t). 2 dt dt We mention in passing that sometimes we consider second order differential equations in the form a(t) y ￿￿ + b(t) y ￿ + c(t) y = f (t). Upon dividing both sides by a(t), we can place this equation in the one above, where p(t) = b(t) , a(t) q (t) = c(t) , a(t) and g (t) = f (t) . a(t) Example. Consider an object of mass m which is under the influence of gravity and air resistance. Let x = x(t) denote the height of the mass at time t. The force due to gravity is m g , and the force due to air resistance is γ v for some constant γ . Newton’s Second Law of Motion states that F = m a is the sum of the forces on the object. This gives the differential equation m d2 x dx = −m g − γ . dt2 dt Hence we have an equation in the form ￿ ￿ d2 x dx = G t, x, where dt2 dt This is a linear second order differential equation. 1 G(t, x1 , x2 ) = −g − γ x2 . m ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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