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Integrating Factors found by Inspection.
1. y(2xy + 1)dx − xdy = 0
2. y(x^4 − y^2)dx + x (x^4 + y^2)dy = 0
3. (x^3y^3 + 1)dx + x^4y^2dy = 0
4. y(x^2y^2 − 1)dx + x(x^2y^2 + 1)dy = 0
5. y(2x + y^2)dx + x(y^2 − x)dy = 0
6. y(3x^3 − x + y)dx + x^2(1 − x^2)dy = 0
7. y(3x^3 − x + y)dx + x^2(1 − x^2)dy = 0
8. y^2(1 − x^2)dx + x(x^2y + 2x + y) = 0
2) Suppose a beam whose one side is free and other side is fixed, ie a cantilever is subjected to a horizontal force ,applied at the free end. Let I=length of the beam ,F=applied force at the free end of the cantilever ,E=modulus of elasticity of the beam ,therefore th moment equation can be written as 𝐸𝐼 𝑑 2𝑦 𝑑𝑥 2 = −𝐹𝑦 − 𝜔 𝑥 2 2 .Solve the differential equation
A 2 kg mass is attached to a spring having spring constant 10 N/m. The mass is placed in a surrounding medium with damping force numerically equal to 8 times the instantaneous velocity. The mass is initially released from rest at 1 meter above the equilibrium position. Find the equation of motion.
Show that the solution y(x) to the equation sqrt(a2-y2)- alna(a+sqrt(a2-y2))+aln(y)+c=0. Is the solution of the differential equation dy/dx= -y/sqrt(a2-y2)
5(b) In a population of lions, the proportionate death rate is 0.55 per year and the
proportionate birth rate is 0.45 per year. Formulate a model of the population. Solve
the model and discuss its long term behavior. Also, find the equilibrium point of the
model.
4. A park has a stable population of birds. Prior to this situation, the birds’ population
increased from an initial low level. When the population of birds was 1000, the
proportionate birth rate was 40% per year and the proportionate death rate was 5% per
year. When the population was 3,000, the proportionate birth rate was 30% and the
proportionate death rate was 10%. Consider the population model under the following
assumptions: (10)
(i) There is no migration and no exploitation.
(ii) The proportionate birth rate is a decreasing linear function of the population.
(iii) The proportionate death rate is an increasing linear function of the population.
Show that
The population grows according to the logistic model.
Find the stable population size.
If the shooting of birds is allowed at the rate of 15% of the population per year, find the
new equilibrium population.
Solve the PDE dz/dx+dz/dy = z^2
dz/dx ×dz/dy -(dz/dy)^2=0 is a non-linear PDE. True or false? Justify.
A bar, 10 cm long with insulated sides, has its ends A and B kept at 20°C and 40°C respectively until steady state conditions prevail. The temperature at A is then suddenly raised to 50°C and at the same instant that at B is lowered to 10°C. Find the subsequent temperature at any point of the bar at any time. (16)
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v) 2e^(x) is a particular integral for y'-y=2e^(x) . True or false
