By L Dresner

This creation to the appliance of Lie's conception to the answer of differential equations comprises labored examples and difficulties. The textual content indicates how Lie's team idea of differential equations has functions to either usual and partial differential equations.

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Thus the assumption that u ( x ) is negative leads to a contradiction. This is enough to prove the ordering. 14~)cannot be solved explicitly, so how shall we proceed? This is an important question because in many applications of Lie's reduction theorem the associated differential equation cannot be integrated in closed form. We must then turn to numerical integration; but it is not yet clear which among the infinitude of integral curves of the associated differential equation is the one to calculate.

13 The integral curves of the differential equation define an envelope. Find a group that leaves the differential equation invariant. Determine from the group the equation of the envelope. 3). 32 First-Order Ordinary Diferential Equations (b) Use the results of part (a) to find a substitution for y that separates the variables in the equation. (c) Use this new variable to determine the coefficients 6 and r] of the infinitesimal transformation of a group G under which the equation is invariant. [Hint: consider Eqs.

C) Use the result of part (a) to determine the new variables u and v and find the transformed differential equation in the variables u and v . 3), find a substitution for u that separates the variables. Thus find a substitution for y that separates the variables in the Bernoulli equation. (b) Use Eqs. 6) to calculate 6 and q for a group that leaves the Bernoulli equation invariant. Tubulution of Differential Eqrtutions 33 (c) Determine relations that the coefficients 6 and q must obey directly from the Bernoulli equation and use them to check your results from part (b).