Science:Math Exam Resources/Courses/MATH101 C/April 2024/Question 08/Solution 1

Since the given differential equation is separable, we can separate the variables and integrate both sides as follows

$${\begin{aligned}&\int e^{y}\;\mathrm {d} y=\int t\;\mathrm {d} t\\&e^{y}={\frac {t^{2}}{2}}+C.\\\end{aligned}}$$

Now, we can solve for $y(t)$ by taking the natural logarithm of both sides

$${\begin{aligned}&y=\log \left({\frac {t^{2}}{2}}+C\right).\\\end{aligned}}$$