Logistic growth in India The population of India was 435 million in 1960 (t=0) and 487 million in 1965 (t=5). The projected population for 2050 is 1.57 billion.
b. Use the solution of the logistic equation and the 2050 projected population to determine the carrying capacity.

A first-order equation Consider the equation t² y′(t) + 2ty(t) = e⁻ᵗ

a. Show that the left side of the equation can be written as the derivative of a single term.

A first-order equation Consider the equation t² y′(t) + 2ty(t) = e⁻ᵗ

c. Find the solution that satisfies the condition y(1) = 0

33–42. Solving initial value problems Solve the following initial value problems.

y'(x) = 4 sec² 2x, y(0) = 8

21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.

y'(t) = t lnt + 1

State the order of the differential equation and indicate if it is linear or nonlinear. 

$y^{{}^{\prime \prime \prime }}+3xy=4\sqrt{x}$

State the order of the differential equation and indicate if it is linear or nonlinear. 

$\frac{d^{2}y}{d{x}^2}-\left(\frac{dy}{dx}\right)(1-x)=2$

State the order of the differential equation and indicate if it is linear or nonlinear. 

${\left(y^{\prime \prime }\right)}^2+6e^t{y}^{\prime }=4t$

State the order of the differential equation and indicate if it is linear or nonlinear. 

$y^{\prime }\bullet y=3e^t$