Differential Equations Answers

Verify that the equations i) z =sqrt (2x + a )+ sqrt(2y + b) and ii)z^2+u=2(1+l ^x)(x+ly) are both complete integrals of the PDEz=1/p+1/q . Also show that the complete integral (ii) is the envelope of one parameter sub-system obtained by taking b=-a/l -μ/1+l in the solution (i)

Expert's answer

Interpret the initial value problem

0

0

0

2

2

2

0, (0) , w

q

b q q q

q

=




+ = =

t= dt

d

dt

d

for any physical situation and hence solve the problem

Solve: z( p − q) = z^2 + (x + y^2)

Two kinds of bacteria are found in a sample of tainted food. It is found that the populations size of type 1,\\(N_1\\) and of type 2, \\(N_2\\) satisfy the equations \\(\\frac{dN_1}{dt}=-k_1N_1, N_1(0)=N_{1,0}\\) and \\(\\frac{dN_2}{dt}=-k_2N_2, N_2(0)=N_{2,0}\\) . Then the population sizes equal \\(N_1=N_2\\) at the following time

Find the integral surface of the partial differential equation

(x y) y p (y x) x q (x y ) z 2 2 2 2 − + − = +

through the curve , 0 2 xz = a y =

solve D.E. y''=1+(y' )^2

apply method of variation of parameters to solve D.E.

a)x^2y''+xy'-y=x^2e^x

solve the D.E. ydouble prime =1+( y prime)^2

Apply the method of variations of parameters to solve the following differential equations:

a) x^2y double prime+xyprime-y=x^2e^x

particle moves in a circle according to the equation \\(\\bar{r}=cos (t^2)\\hat{i} +sin(t^2)\\hat{j}\\). The magnitude of the normal component of the acceleration at time \\(t\\) is