Solution of Differential Equation

Once a differential equation is available, the next question is what it means to SOLVE it, and the simplest systematic technique for doing so — separation of variables.

GENERAL AND PARTICULAR SOLUTIONS. A differential equation typically has infinitely many solution functions, not just one. The textbook illustrates this by checking that both y = a sin x and y = b cos x satisfy the same second-order differential equation individually, and that y = a sin x + b cos x — carrying BOTH arbitrary constants a and b at once — also satisfies it. Here y = a sin x and y = b cos x are called PARTICULAR solutions, while y = a sin x + b cos x, which describes every possible solution as a and b range over all values, is the GENERAL solution: a solution containing as many independent arbitrary constants as the order of the equation. Any solution obtained from the general solution by assigning specific values to those arbitrary constants (typically forced by extra given data, such as an initial condition) is called a particular solution. Because of the arbitrary constants, a differential equation genuinely has infinitely many solutions.

Two solved examples verify that a given expression is indeed a solution of a stated equation, by differentiating the expression and substituting: (Ex.1) y·sec x = tan x + c is shown, by differentiating implicitly, to satisfy dy/dx + y·tan x = sec x. (Ex.2) y = log x + c is shown, by differentiating twice, to satisfy x(d²y/dx²) + dy/dx = 0.

METHOD OF SEPARATION OF VARIABLES. Considering the illustrative equation dy/dx = x²y + y = y(x²+1): dividing both sides by y gives (1/y)(dy/dx) = x²+1, which can be rewritten treating x and y both as variables: dy/y = (x²+1)dx. Integrating the left side with respect to y and the right side with respect to x gives log y = x³/3 + x + c. This process — separating an equation into a pure function of x times dx on one side and a pure function of y times dy on the other — is the method of separation of variables. In general, whenever a given differential equation can be manipulated into the form f(x)dx = g(y)dy, this method applies directly: integrate both sides independently and combine the resulting two constants of integration into one.

Three further pairs of solved examples work through this method and its natural extensions:

Ex.1(i) dy/dx = x√(25−x²): separating gives dy = x√(25−x²)dx; substituting t = 25−x² (so x dx = −dt/2) turns the right side into −(1/2)√t dt, and integrating gives 2y + (2/3)t^(3/2) = c₁, i.e. 6y + 2(25−x²)^(3/2) = c.

Ex.1(ii) dx/dt = (x log x)/t: separating gives dx/(x log x) = dt/t; the left integral (with u = log x) gives log(log x), so log(log x) = log t + log c, i.e. log x = ct, i.e. x = e^(ct).

Ex.2(i) Particular solution of dy/dx = e^(2y)cos x when x=π/6, y=0: separating gives e^(−2y)dy = cos x·dx; integrating gives e^(−2y)/(−2) = sin x + c; using the initial condition gives c = −1; so the particular solution is e^(2y)(2 sin x − 2) + 1 = 0. …