Задача 80603 X2+y2=lny/x...
Условие
математика
18
Решение
★
[m]x^2+y^2=\frac{lny}{x} [/m]
[m]2x+2y\cdot y'=\frac{\frac{y'}{y} \cdot x-lny}{x^2} \\\\
2x+2y\cdot y'-\frac{y'}{xy} +\frac{lny}{x^2}=0\\\\
2yy'-\frac{y'}{xy} =-2x-\frac{lny}{x^2} \\\\
y'(2y-\frac{1}{xy})=-\frac{2x^3+lny}{x^2} \\\\
y'=(-\frac{2x^3+lny}{x^2} ):(\frac{2xy^2-1}{xy} )\\\\
y'=-\frac{2x^4y+xy lny}{2x^3y^2-x^2} [/m]
[m]y'=-\frac{2x^3y+ylny}{2x^2y^2-x} [/m]