This comes from the book Differential Equations and Linear Algebra 4th edition.

Solve the initial value problem yy' + a: = V'mg + y\" with y(2) = 1/?E. a. To solve this, we should use the substitution 1: = 9', help (formulas) w 2 imam Enter derivatives using prime notation [e.g., you would enter 3," for g). b. After the substitution from the previous part, we obtain the following linear differential equation in 3,11,,'11 '. #1, help (equations) :3. The solution to the original initial value problem is described by the following equation in z, y. Solve the initial value Problem 13' = (a: + y 4)2 with y(0) = 0. a. To solve this, we should use the substitution 1:. = .92, ' help {formulas} 'u ' = it, help (formulas) dy ). Enter derivatives using prime notation {e.g., you would enter *9\" for E b. tter the substitution from the previous part, we obtain the following differential equation in z,u,u'. c. The solution to the original initial value problem is described by the following equation in Ly. ' if, help (equations) A Bernoulli differential equation is one of the form dx dy + P(x)y = Q(x)y". Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = yl-" transforms the Bernoulli equation into the linear equation du + (1 -n)P(x)u = (1 -n)Q(x). Use an appropriate substitution to solve the equation 3 y -y = and find the solution that satisfies y(1) = 1. y(x) =A Bernoulli differential equation is one of the form dy + P(x)y = Q(x)y". Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = yl-" transforms the Bernoulli equation into the linear equation du do + (1 -n)P(x)u = (1 -n)Q(x). Use an appropriate substitution to solve the equation ry'ty=-3xy', and find the solution that satisfies y(1) = 2. y(x) =