Question

Find the particular solution of the differential equation that satisfies the initial conditions.

Answer 1

`\begin{gathered} f^(\prime\prime)(x)=-(4)/((x-1)^2)-2 \\ u=x-1 \\ f^(\prime)(x)=\int_(x>1)(-(4)/((x-1)^2)-2)\cdot dx=-4\int_(u>0)u^(-2)\cdot du-2\int_(x>1)dx+c \\ f^(\prime)(x)=(4)/(u)-2x \\ f^(\prime)(x)=(4)/(x-1)-2x+c \\ f^(\prime)(x)=(-2x^2+2x+4)/(x-1)+c \\ f^(\prime)(2)=0 \\ c=-((-2\cdot2^2+2\cdot2+4))/(2-1) \\ c=0 \\ \begin{equation*} f^(\prime)(x)=(-2x^2+2x+4)/(x-1) \end{equation*} \\ f(x)=\int_(x>1)(-2x^(2)+2x+4)/(x-1)\cdot dx \\ f(x)=-2\int_(x>1)(x^2)/(x-1)dx+2\int_(x>1)(x)/(x-1)dx+4\int(1)/(x-1)dx \\ f(x)=-(x^2+2x+2\ln|x-1|-3)+2(x-1+\ln|x-1|)+2\ln|x-1|+c \\ f(x)=-x^2+2\ln|x-1|+1+c \\ f(2)=3 \\ -2^2+2\ln|2-1|+1+c=3 \\ c=6 \\ \therefore f(x)=-x^2+2\ln|x-1|+7 \end{gathered}`