Question

Decide whether or not the following equations are linear:

(a) d^2/dx^2y(x) = -8(y(x))^2 is a

- linear equation

- non-linear equation

(b) d/dx(yx) + sin(4y) = 0 is a

- linear equation

- non-linear equation

(c) sin(8x)*d/dxy(x) + y(x) = 7x is a

- linear equation

- non-linear equation

(d) d/dtx(t) +3x = -8t^3 is a

- linear equation

- non-linear equation

(e) sin(5y(x))d/dxy(x) + y(x) = 9x is a

- linear equation

- non-linear equation

Answer #1

(x+1)y'=y-1 2) dx+(x/y+(e)?)dy=0 3)ty'+2y=sint 4) y"-4y=-3x²e3x
5) y"-y-2y=1/sinx 6)2x2y"+xy'-2y=0 ea)y=x' b) x=0
2dx/dt-2dy/dt-3x=t; 2

Given parametric equations below, find d^2y/dx^2 and determine
the intervals on which the graph of the curve is concave up or
concave down.
(a) x = t^2 , y = t^3−3t
(b) x = cos(t), y = sin(2t)

Find d^2y/dx^2 . x = t^3 − 7, y = t − t^2
d^2y/dx^2=?

(differential equations): solve for x(t) and y(t)
2x' + x - (5y' +4y)=0
3x'-2x-(4y'-y)=0
note: Prime denotes d/dt

d^2/dx^2*y(x) +64y(x) = 0, select all solutions.
a) y(x) = C1sin(9x)
b) y(x) =9cos(8x)
c) y(x) = C2cos(2x)
d) y(x) = 9sin(9x) + 5cos(9x)
e) y(x) = 9cos(2x)

For each of the following equations, find the general solution
(perferably using imaginary numbers):
y′′ −2y′ −3y=3e2t;
y′′ +2y′ =3+sin(2t);
x2y′′ +7xy′ +5y=x;
x2y′′ +xy′ +4y=sin(lnx)+sin(2lnx);
y′′ +y=tcost.

Non homogeneous eq w constant; undetermined coefficients
Find the general solution:
1) y" + 4y' + 4y = xe^−x.
2) y" + 2y' + 5y = e^2x cos x.
Determine a suitable form for a particular solution z = z(x) of
the given equations
1) y" + 2y' = 2x + x^2e^−3x + sin 2x.
2) y" − 5y' + 6y = 2e^2x cos x − 3xe^3x + 5.
3) y" + 5y' + 6y = 2e^2x cos x −...

solve for x(t) and y(t):
x'=-3x+2y;
y'=-3x+4y
x(0)=0,y(0)=2

Solve the following differential equations
y''-4y'+4y=(x+1)e2x (Use Wronskian)
y''+(y')2+1=0 (non linear second order equation)

1- Find the solution of the following equations. For each
equation, 2- determine the type of the category that the equation
belongs to.
1. y/x cos y/x dx − ( x/y sin y/x + cos y/x ) dy = 0
2. x(1 − y^2 )dx + y(8 − x^2 )dy = 0
3. (x^2 − x + y^2 )dx − (e^y − 2xy)dy = 0
4. 2x sin 3ydx + 3x^2 cos 3ydy = 0
5. (x ln x −...

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