4.3 Separable equations  (Page 4/8)

$\frac{dy}{dt}=y+1$

$\frac{dy}{dt}=y-1$

$\frac{dy}{dt}=y+1$

$\frac{dy}{dt}=\text{−}y-1$

$x^{2}y\prime =\left(x+1\right)y$

$y\prime =\text{tan}\left(y\right)x$

$y\prime =2x{y}^2$

$\frac{dy}{dt}=y\text{cos}\left(3t+2\right)$

$2x\frac{dy}{dx}=y^2$

$y\prime =e^y{x}^2$

$\left(1+x\right)y\prime =\left(x+2\right)\left(y-1\right)$

$\frac{dx}{dt}=3t^2\left(x^2+4\right)$

$t\frac{dy}{dt}=\sqrt{1-y^2}$

$y\prime =e^x{e}^y$

$y\prime =e^{y-x},y(0)=0$

$y\prime =y^2(x+1),y(0)=2$

$\frac{dy}{dx}=y^{3}x{e}^{x^2},y(0)=1$

$\frac{dy}{dt}=y^2{e}^x\text{sin}(3x),y(0)=1$

$y\prime =\frac{x}{{\text{sech}}^{2}y},y(0)=0$

$y\prime =2xy(1+2y),y(0)=\mathrm{-1}$

$\frac{dx}{dt}=\text{ln}(t)\sqrt{1-x^2},x(0)=0$

$y\prime =3x^2(y^2+4),y(0)=0$

$y\prime =e^y{5}^x,y(0)=\text{ln}(\text{ln}(5))$

$y\prime =\mathrm{-2}x\text{tan}(y),y(0)=\frac{\pi }{2}$

[T] $y\prime =1-2y$

[T] $y\prime =y^2{x}^3$

[T] $y\prime =y^3{e}^x$

[T] $y\prime =e^y$

[T] $y\prime =y\text{ln}(x)$

Most drugs in the bloodstream decay according to the equation $y\prime =cy,$ where $y$ is the concentration of the drug in the bloodstream. If the half-life of a drug is $2$ hours, what fraction of the initial dose remains after $6$ hours?

A drug is administered intravenously to a patient at a rate $r$ mg/h and is cleared from the body at a rate proportional to the amount of drug still present in the body, $d$ Set up and solve the differential equation, assuming there is no drug initially present in the body.

[T] How often should a drug be taken if its dose is $3$ mg, it is cleared at a rate $c=0.1$ mg/h, and $1$ mg is required to be in the bloodstream at all times?

A tank contains $1$ kilogram of salt dissolved in $100$ liters of water. A salt solution of $0.1$ kg salt/L is pumped into the tank at a rate of $2$ L/min and is drained at the same rate. Solve for the salt concentration at time $t.$ Assume the tank is well mixed.

A tank containing $10$ kilograms of salt dissolved in $1000$ liters of water has two salt solutions pumped in. The first solution of $0.2$ kg salt/L is pumped in at a rate of $20$ L/min and the second solution of $0.05$ kg salt/L is pumped in at a rate of $5$ L/min. The tank drains at $25$ L/min. Assume the tank is well mixed. Solve for the salt concentration at time $t.$