Indiana State University Differential Equations Please solve the question in the picture that i attached. I need the answer within 30 minutes. Thank you. I

Indiana State University Differential Equations Please solve the question in the picture that i attached. I need the answer within 30 minutes. Thank you. I need detailed steps. Question 1
100 pts
1. Let P(x) denote the solution to the initial value problem
dy
dc
= x – y, y(0) = 1.
3.x2
da2
a) Show that “(x) = 1 – *'(x)
b) Show that “(x) = 1 – 2 – (x).
2. Determine the values of m for which °(x) = x” is a solution to the differential equation
day
+ 112 – 3y = 0 .
3. Show that the equation y- dy + yecos i sin xdx = 0 is separable. Hence solve the initial
value problem ydy + yecos I sin xdx = 0, y(0) = 1
4. Find the general solution to 23 = X COSI, 2 > 0.
1 dy
I do
5. Consider the equation
(y2 + 2xy)dx – x? dy = 0
a) Show that this equation is not exact.
y 2 yields a new equation that is
b) Show that multiplying both sides of the equation by
exact.
c) Use the solution of the resulting exact equation to solve the original equation.
d) Were any solutions lost/gain in the process? (Hint: check if y = c is a solution to the original
equation but not the new equation and vice versa. Determine this constant c).
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