DIFF EQ TRIAL

Of course, trial and error is not the best way to solve an equation, but if you are lucky (or practiced) enough to actually discover a solution by inspection, you should be rewarded.

For exmple, you might discoer that the simple function yР’В = xР’В is a solution of the equation Type 3: SecondРІР‚С’order homogeneous linear equations where one (nonzer) solution is known.Р’В Sometimes it is possible to determine a solution of a secondРІР‚С’order differential equation by inspection, which usually amounts to successful trial and error with a few particularly simple functions. Therefore, the complete solution of the given differential equation is Which gives the general solution, expressed implicitly as follows: Now, since wР’В = dy/dx, this last result becomes The second statement is a separable equation, and its solution proceeds as follows: The statement wР’В = 0 means yРІР‚Р† = 0, and thus yР’В = cР’В is a solution for any constant c. The substitutions yРІР‚Р† = wР’В and yРІР‚С– = w( dw/dy) tranform this secondРІР‚С’order equation for yР’В into the following firstРІР‚С’order equation for w: Once wР’В is determined, integrate to find y.Р”В®xample 4: Solve the differential equation This substitution, along with yРІР‚Р† = w, will reduce a Type 2 equation to a firstРІР‚С’order equation for w. This is accomplished using the chain rule:

But instead of simply writing yРІР‚С– as wРІР‚Р†, the trick here is to express yРІР‚С– in terms of a first derivative with respect to y. The method for reducing the order of these secondРІР‚С’order equations begins with the same substitution as for Type 1 equations, namely, replacing yРІР‚Р† by w. The defining characteristic is this: The independent variable, x, does not explicitly appear in the equation. Type 2: SecondРІР‚С’order nonlinear equations with the independent variable missing.Р’В Here's an example of such an equation: Thus the solution of this IVP (at least for xР’В > РІв‚¬вЂ™1) is Now apply the initial conditions to determine the constants c 1Р’В and c 2.Р”В«ecause c 1Р’В = 1, the first condition then implies c 2Р’В = 1 also. The differential equation is transformed into Letting c 1Р’В = РІвЂ¦вЂњ c 1, the general solution can be writtenР”В®xample 3: Sketch the solution of the IVPР”Р„lthough this equation is nonlinear, the substitutions yРІР‚Р† = wР’В and yРІР‚С– = wРІР‚Р† will still reduce this to a firstРІР‚С’order equation, since the variable yР’В does not explicitly appear. Which is used to multiply both sides of the equation, yielding (This particular differential equation could also have been solved by applying the method for solving secondРІР‚С’order linear equations with constant coefficientes.)Р”В®xample 2: Solve the differential equationР”Р„gain, the dependent variable yР’В is missing from this secondРІР‚С’order equation, so its order will be reduced by making the substitutions yРІР‚Р† = wР’В and yРІР‚С– = wРІР‚Р†: Referring to Theorem B, note that this solution implies that yР’В = c 1 e РІв‚¬вЂ™ x+ c 2Р’В is the general solution of the corresponding homogeneous equation and that yР’В = Р’Р… x 2Р’В РІР‚вЂњ xР’В is a particular solution of the nonhomogeneous equation. Now, to give the solution yР’В of the original secondРІР‚С’order equation, integrate:

Applying the method for solving such equations, the integrating factor is first determined,Р”Р„nd then used to multiply both sides of the equation, yielding These substitutions transform the given secondРІР‚С’order equation into the firstРІР‚С’order equation Since the dependent variable yР’В is missing, let yРІР‚Р† = wР’В and yРІР‚С– = wРІР‚Р†. Solve for the function w then integrate it to recover y.Р”В®xample 1: Solve the differential equation yРІР‚Р† + yРІР‚С– = w. This substitution obviously implies yРІР‚С– = wРІР‚Р†, and the original equation becomes a firstРІР‚С’order equation for w. This type of secondРІР‚С’order equation is easily reduced to a firstРІР‚С’order equation by the transformation The defining characteristic is this: The dependent variable, y, does not explicitly appear in the equation. Type 1: SecondРІР‚С’order equations with the dependent variable missing.Р’В Examples of such equations include Type 3: SecondРІР‚С’order homogeneous linear equations where one (nonzero) solution is known Type 2: SecondРІР‚С’order nonlinear equations with the independent variable missing Type 1: SecondРІР‚С’order equations with the dependent variable missing

Diff eq