Standard +0.3 This is a standard separable differential equation with straightforward integration and boundary conditions. While it requires multiple steps (separation, integration, applying two conditions, and expressing in the required form), each step follows routine procedures taught in any differential equations course. The algebra is clean and the final form is explicitly requested, making this slightly easier than average.
13 Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - k ( y - 10 )\), where \(k\) is a constant, given that \(y = 70\) when \(x = 0\) and \(y = 40\) when \(x = 1\). Express your answer in the form \(y = a + b \left( \frac { 1 } { 2 } \right) ^ { x }\) where \(a\) and \(b\) are constants to be found. [0pt]
[10]
13 Solve the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } = - k ( y - 10 )$, where $k$ is a constant, given that $y = 70$ when $x = 0$ and $y = 40$ when $x = 1$. Express your answer in the form $y = a + b \left( \frac { 1 } { 2 } \right) ^ { x }$ where $a$ and $b$ are constants to be found.\\[0pt]
[10]
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2013 Q13 [10]}}