2 questions from 2026
Let \( y:(-\infty,\infty)\to(0,\infty) \) be the solution of the differential equation \( \frac{dy}{dx}=\frac{e^{5x}y^{3}+y^{3}}{e^{x}+e^{x}y^{4}} \), satisfying \( y(0)=\frac{1}{\sqrt{2}} \). Then the value of \( y(\log_{e}2) \) is:
Let \( y=f(x) \) be the real valued function defined on the interval \( (0,\infty) \), satisfying \( y(1)=0 \) and the differential equation \( x\,\dfrac{dy}{dx}=y-x^3 \). Then which of the following statements is (are) TRUE?
