Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

GRYTCZUK, K., General solution of the differential equation ...

General solution of the differential equation y"(x) - (y'{x))

2 + = 0 KRYSTYNA GRYTCZUK Abstract. In this note we prove that the general solution of the differential equation y"(x)~ (y'(x)) 2+x

2e

y(x

)=0, x>0 is the function y(x)=- In W(x), where W(x)=Y2x

i+Ax-\B and A,B are arbitrary constants. 1. Introduction In this note we prove that the general solution of the differential equation (1) y"{x) - (y'(x))

2 + x

2e

y{x

) = 0, x > 0 is the function (2) y(x) = -\nW(x), where W(x) = ^x

4 + Ax + B and A, B are arbitrary constants. First, we note that such type of differential equations as (1) are difficult to solve. For example, E. Y. RODIN (see [1], p. 474, Unsolved problems, SIAM 81-17) posed the following problem. Find the general solution of the differential equation: (3) y"(x) + x

2e

y{x

) = 0, x > 0. We prove that (1) has general solution given by (2), however we can't fjid the general solution of (3). 2. The Result We prove the following theorem: Theorem. The general solution of the differential equation (1) is the function y(x) = ln(^x

2+Ax+ where A,B are arbitrary constants.