Question

Find all solutions to the differential equations.

(a) x^2 yy' = (y^2 − 1)^(3/2)

(b) y' = 6xe^(x−y)

(c) y' = (2x − 1)(y + 1)

(d) (y^2 − 1) dy/dx = 4xy^2

Leave your answer as an implicit solution

Answer #1

Solve using basics of differential equations

3. Consider the differential equation: x dy/dx = y^2 − y
(a) Find all solutions to the differential equation.
(b) Find the solution that contains the point (−1,1)
(c) Find the solution that contains the point (−2,0)
(d) Find the solution that contains the point (1/2,1/2)
(e) Find the solution that contains the point (2,1/4)

(61). (Bernoulli’s Equation): Find the general solution of the
following first-order differential equations:(a) x(dy/dx)+y=
y^2+ln(x) (b) (1/y^2)(dy/dx)+(1/xy)=1

1- Solve Ordinary Differential Equations
A) (xˆ2 + 1) y '+ 4xy = x
B) y '+ ((2x + 1) / x) y = exp (-2x)
C) y '+ y = sen (x)
D) y'-2xy = x

(* Problem 3 *)
(* Consider differential equations of the form a(x) + b(x)dy
/dx=0 *) \
(* Use mathematica to determin if they are in Exact form or not.
If they are, use CountourPlot to graph the different solution
curves 3.a 3x^2+y + (x+3y^2)dy /dx=0 3.b cos(x) + sin(x) dy /dx=0
3.c y e^xy+ x e^xydy/dx=0

Find a particular solution to each of the following differential
equations with the given initial condition:
A) dy/dx=ysinx/1+y^2, y(0)=1
B)dy/dx=xy ln x, y(1)=3
C)xy(1+x^2) dy/dx=(1+y^2), y(1)=0

Consider x^2 +sin(y)=4xy^2 +1
a.)Use Implicit differentiation to find dy/dx
b.) find an equation tangent of the line to the curve x^2
+sin(y)=4xy^2 +1 at (1,0)

(Part 1) Find all of the solutions of the given differential
equations:
a.) y' = -2y (answer should be y = -(1 / ln2) * ln(t
* ln(2) + c))
(Part 2) Find the solution of the IVP:
b.) y' = -2y3, y(0) = 0
c.) y' = 1 + cos(y), y(0) = pi / 2 (answer should be y(t) =
2arctan(1 + t))
d.) y' = sqrt(1 - y2), y(0) = 0 (Hint: y' > 0)
Please show work!

3. Find the general solution to the differential equation:
(x^2 + 1/( x + y) + y cos(xy)) dx + (y ^2 + 1 / (x + y) + x
cos(xy)) dy = 0

Solve the following differential equations.
a.) dy/dx+2xy=x, y(0)=2
b.) ?^2(dy/dx)−?y=−y^2

Find the solution of the following differential
equation:
(?^3 y/dx^3)-7(d^2 y/dx^2)+10(dy/dx)=e^2x sinx

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