Section 9.3: Separable differential equations   Lecture Video

1. Find the general solution of the following differential equation.

(a) \(\displaystyle\frac{dy}{dx}=6x\sqrt{y}\)  solution

(b) \(\displaystyle \frac{dy}{dx}= \frac{1+\sin(x)}{y}\)  solution

(c) \(\displaystyle \frac{dy}{dx}= \frac{x+1}{xy}\)  solution

(d) \(\displaystyle \frac{d u}{d t}=\frac{2+t^{4}}{u^3 t^{2}+u^{4} t^{2}}\)  solution

2. Find the solution of the differential equation that satisfies the given initial condition.

(a) \(\displaystyle \frac{dy}{dx}=2y, \: \: y(1)= e\)   solution

(b) \(\displaystyle \frac{dy}{dx}= xe^y, \: \: y(0)= 0\)  solution

(c) \(\displaystyle \frac{dy}{dx}=\frac{3}{1+x^2}, \:\: y(1)=\pi\)   solution

3. Find an equation of the curve that passes through the point \((1, 3)\) and whose slope at \((x, y)\) is \(\dfrac{x}{y}\).  solution

4. The differential equation below models the temperature of an \(93^0\)C cup of coffee in a \(17^0\)C room, where it is known that the coffee cools at a rate of \(1^0\)C per minute when its temperature is \(67^0\)C. Solve the differential equation to find an expression for the temperature of the coffee at time t. (Let \(y\) be the temperature of the cup of coffee in \(^°\)C, and let \(t\) be the time in minutes, with \(t=0\) corresponding to the time when the temperature was \(93^0\)C.)   \(\frac{dy}{dt}=-\frac{1}{50}(y-17)\)     Solution