Solve the given initial value problem for y = f (x). $$\frac { d y } { d x } = \frac { x - 5 } { x ^ { 2 } - 10 x + 8 }$$ where y = 6 when x = 2
A) $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right|$
B) $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right| + 6$
C) $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right| - \frac { 1 } { 2 } \ln 8 + 6$
D) $y = \frac { 1 } { 2 } \ln \left| x ^ { 2 } - 10 x + 8 \right| - \frac { 1 } { 2 } \ln 8$

Question

Quizplus · Accepted Answer

To solve the given initial value problem, we integrate the given differential equation. The integral of the right-hand side involves a simple u-substitution with $u = x^2 - 10x + 8$, leading to $\frac{1}{2}\ln|u| + C$ as the general solution. Substituting back for $u$ gives $\frac{1}{2}\ln|x^2 - 10x + 8| + C$. Using the initial condition $y(2) = 6$, we find $C = -\frac{1}{2}\ln 8 + 6$, resulting in the specific solution $y = \frac{1}{2}\ln|x^2 - 10x + 8| - \frac{1}{2}\ln 8 + 6$.

Solve the Given Initial Value Problem for Y = F dydx=x−5x2−10x+8\frac { d y } { d x } = \frac { x - 5 } { x ^ { 2 } - 10 x + 8 }dxdy​=x2−10x+8x−5​

Solve the Given Initial Value Problem for Y = F $\frac { d y } { d x } = \frac { x - 5 } { x ^ { 2 } - 10 x + 8 }$