Question

Solve each differential equation.

a) dy/dx= x²y²−x²+4y2−4
b) (x-1)dy/dx - xy=e⁴x
c) (7x-3y)dx+(6y-3x)dy=0
Solve the following initial value problem
1) (3x² + y-2)dx +(x+2y)dy=0 y(2)=3
2)show that 5xy² + sin(y)= sin(x² +1) is an implicite solution to the differential equation: dy/dx=2xcos(x²+1)-5y²/10xy+cos(y)
3) find value for k for which y= eᵏˣ is a solution of the differential equation y"-11y'+28y=0
4)A tank contains 480 gallons of water in which 60 lbs of salt are dissolved. A saline solution containing 0.5 lbs of salt per gallon is pumped into the tank at the rate of 2 gallons per minute. The well-mixed solution is pumped out at the rate of 4 gallons per minute. Set up an initial value problem which can be solved for the amount A of salt in the tank at time t
5)Consider the following differential equation:
sin(x) d³y/dx³-x² dy/dx+y= lnx
(a) Is the equation linear ornonlinear?
(b) Is it a partial or ordinary differential equation?
(c) What is the order of the equation?
6) Verify that
y= x² ln(x) is a solution of
x² y"' + 2xy"- 3y'+ (1/x) y= 5x- xln(x)
on the interval (0, inf)
8)Determine if the following differential equation is homogeneous or not.

3x² y dx + (x² + y²)dy=0

Answer 1

a) Separate variables and integrate to solve. b) Use an integrating factor and integrate to solve. c) Rearrange and integrate to solve.

Step-by-step explanation:

a) To solve the differential equation dy/dx = x²y²−x²+4y²−4, we can separate variables and integrate both sides of the equation. Rearranging, we get dy/(y²+4y²-4) = x²dx. Now, integrate both sides: ∫(1/(y²+4y²-4))dy = ∫x²dx. Evaluate the integrals and solve for y to find the solution.

b) The given differential equation is linear. To solve it, we can use an integrating factor. Multiply both sides of the equation by (x-1) to get (x-1)dy/dx - xy = e^(4x)(x-1). Now, we can view the left side as the derivative of (x-1)y, and integrate both sides to find the solution.

c) To solve the equation (7x-3y)dx + (6y-3x)dy = 0, we can rearrange the equation to get (7x-3y)/(6y-3x)dx = -dy. Now, separate variables and integrate both sides to find the solution.