# On a class of non-linear differential equations with exact solutions

## The present paper deals with a class of non-linear ordinary second-order differential equations with exact solutions. A procedure for finding the general exact solution based on a known particular one is derived. For illustration solutions of some non-linear equations occured in many problems of solid mechanics are considered. | Vietnam Journal of Mechanics, VAST, Vol. 34, No. 1 (2012), pp. 7 – 17 ON A CLASS OF NON-LINEAR DIFFERENTIAL EQUATIONS WITH EXACT SOLUTIONS Dao Huy Bich1 , Nguyen Dang Bich2 1 Hanoi University of Science, VNU 2 Institute for Building Science and Technology Abstract. The present paper deals with a class of non-linear ordinary second-order differential equations with exact solutions. A procedure for finding the general exact solution based on a known particular one is derived. For illustration solutions of some non-linear equations occured in many problems of solid mechanics are considered. Key words: Non-linear differential equation, general exact solution, varying coefficient iteration method, procedure for finding exact solution. 1. INTRODUCTION Generally for seeking an exact solution of a non-linear differential equation it is necessary to find an appropriate transformation deriving the non-linear equation to a linear one, but finding of such transformation is very complicated. In fact many problems of solid mechanics reduce to different types of non-linear differential equations, solutions of which can demonstrate specific effects, however through only exact solutions these effects can be observed profoundly. Hence finding exact solutions becomes very important in researching non-linear mechanical problems. The present paper introduces an idea and a procedure to find general exact solutions of a class of non-linear ordinary second-order differential equations based on known particular solutions. If a particular exact solution is known, then a general exact solution can be found, but for a received approximate particular one, a general solution may be obtained approximately with desired accuracy by the varying coefficient iteration method. 2. IDEA AND PROCEDURE FOR FINDING EXACT SOLUTION Consider a non-linear second-order differential equation 2 d x˙ b x˙ b a1 + a2 x + a3 + + a1 + a2 x + a3 + + dt x+d x+d x+d x+d 2 d x˙ b x˙ b + + b2 x + b3 + + b2

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