The equations that allow lower-order

Some second-order equations can be reduced to first-order equations, rendering them susceptible to the simple methods of solving equations of the first order. The following are three particular types of such second-order equations:

Type 1: Second-order equations with the dependent variable missing

Type 2: Second-order nonlinear equations with the independent variable missing

Type 3: Second-order homogeneous linear equations where one (nonzero) solution is known

Type 1: Second-order equations with the dependent variable missing. Examples of such equations include and .

The defining characteristic is this: The dependent variable, y, does not explicitly appear in the equation. This type of second-order equation is easily reduced to a first-order equation by the transformation . This substitution obviously implies y ″ = w ′, and the original equation becomes a first-order equation for w. Solve for the function w; then integrate it to recover y.

Example 7. Solve the differential equation y ′ + y ″ = x.

Since the dependent variable y is missing, let y ′ = w and y ″ = w ′. These substitutions transform the given second-order equation into the first-order equation which is in standard form. Applying the method for solving such equations, the integrating factor is first determined, and .

Now, to give the solution y of the original second-order equation, integrate: . Note that this solution implies that y = c ₁ e ^{− x} + c ₂ is the general solution of the corresponding homogeneous equation and that y = ½ x ² – x is a particular solution of the nonhomogeneous equation. (This particular differential equation could also have been solved by applying the method for solving second-order linear equations with constant coefficients.)

Type 2: Second-order nonlinear equations with the independent variable missing. Here's an example of such an equation: .

The defining characteristic is this: The independent variable, x, does not explicitly appear in the equation.

The method for reducing the order of these second-order equations begins with the same substitution as for Type 1 equations, namely, replacing y ′ by w. But instead of simply writing y ″ as w ′, the trick here is to express y ″ in terms of a first derivative with respect to y. This is accomplished using the chain rule: .

Therefore, . This substitution, along with y ′ = w, will reduce a Type 2 equation to a first-order equation for w. Once w is determined, integrate to find y.

Example 8. Solve the differential equation .

The substitutions y ′ = w and y ″ = w (dw/dy) tranform this second-order equation for y into the following first-order equation for w:

, .

Therefore, .

The statement w = 0 means y ′ = 0, and thus y = c is a solution for any constant c. The second statement is a separable equation, and its solution proceeds as follows: .

Now, since w = dy/dx, this last result becomes , ,
which gives the general solution, expressed implicitly as follows: .

Therefore, the complete solution of the given differential equation is .

Type 3: Second-order homogeneous linear equations where one (nonzer) solution is known. Sometimes it is possible to determine a solution of a second-order differential equation by inspection, which usually amounts to successful trial and error with a few particularly simple functions. For example, you might discover that the simple function y = x is a solution of the equation or that y = e^x satisfies the equation .

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.

If one (nonzero) solution of a homogeneous second-order equation is known, there is a straightforward process for determining a second, linearly independent solution, which can then be combined wit the first one to give the general solution. Let y ₁denote the function you know is a solution. Then let y = y ₁ v (x), where v is a function (as yet unknown). Substitute y = y ₁ v into the differential equation and derive a second-order equation for v. This will turn out to be Type 1 equation for v (because the dependent variable, v, will not explicitly appear). Use the technique described earlier to solve for the function v; then substitute into the expression y = y ₁ v to give the desired second solution.

Let's consider three types of differential equations of the n-th order, allowing reduction of order.

The equation of the form , where f(x) is a given function. We obtain the general solution of this equation making the sequence of n integrations, with each such integration will appear a new arbitrary constant.

The equation , The equation does not contain a clear y and lower derivatives up to order k-1 inclusive, allows reduction of order by k units.For this purpose, we introduce a new unknown function . Then and equation for z is order n-k: . If you found a general solution of this equation then we have the equation: . Integrating it, we find the general solution of equation (*).

The equation does not contain explicitly the independent variable. Here the order of the equation is reduced to the unit by replacing the two variables. As a new unknown function we choose , as well as the new independent variable is у.

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