\frac{d^2y}{dx^2}=\frac{dt}{dx}\\\\ \frac{dt}{dx}=\frac{xt}{x^2-1} \ \ \mid \div t \\\\\frac{\frac{dt}{dx} }{t}=-\frac{x}{x^2-1}\\\\ \int \frac{\frac{dt}{dx} }{t}dx=\int-\frac{x}{x^2-1}dx\\\\ln(t)=-\frac{1}{2}ln(x^2-1)+C_1\\\\t=\frac{e^{C_1}}{\sqrt{x^2-1}}=\frac{C_1}{\sqrt{x^2-1}}\\\\\frac{dy}{dx}=t=>\frac{dy}{dx}=\frac{C_1}{\sqrt{x^2-1}}=\int\frac{C_1}{\sqrt{x^2-1}}dx=C_1*ln(x+\sqrt{x^2-1})+C_2" alt="(1-x^2)*y''=xy'\\\\(-x^2+1)*\frac{d^2y}{dx^2}=x\frac{dy}{dx}\\\\ \frac{dy}{dx}=t=>\frac{d^2y}{dx^2}=\frac{dt}{dx}\\\\ \frac{dt}{dx}=\frac{xt}{x^2-1} \ \ \mid \div t \\\\\frac{\frac{dt}{dx} }{t}=-\frac{x}{x^2-1}\\\\ \int \frac{\frac{dt}{dx} }{t}dx=\int-\frac{x}{x^2-1}dx\\\\ln(t)=-\frac{1}{2}ln(x^2-1)+C_1\\\\t=\frac{e^{C_1}}{\sqrt{x^2-1}}=\frac{C_1}{\sqrt{x^2-1}}\\\\\frac{dy}{dx}=t=>\frac{dy}{dx}=\frac{C_1}{\sqrt{x^2-1}}=\int\frac{C_1}{\sqrt{x^2-1}}dx=C_1*ln(x+\sqrt{x^2-1})+C_2" align="absmiddle" class="latex-formula">