Using the method introduced in Question 26, find the solutions of the following initial value problems Data from Question 26 Show that by making the substitution Show that the solution of this equation is v 1 2x 2 C and hence find x(t) This technique is a standard method for solving second order di...

a b c d Hence xp Let v Then This is a separable equation H ...

Using the method introduced in Question 26, find the solutions of the following initial-value problems: Data from

Question:

Using the method introduced in Question 26, find the solutions of the following initial-value problems:

dx (a) x- = dt d.x dt 2 x(0) = 4, dx dt -(0)=1

(b) dx dt (d) dx dt d.x dx e, x(1)=1,(1) = 0 dt dt (c) = xdx dt dx 1 / dx + dt 2 dt x(0) = 2, d.x -(0) dt =

Data from Question 26

Show that by making the substitution

d.x dt and noting that dx du dt dt the equation dx dt may be expressed as v= du dx d.x -X = || ax=a dt du dx

Show that the solution of this equation is v = 1/2x² + C and hence find x(t). This technique is a standard method for solving second-order differential equations in which the independent variable does not appear explicitly. Apply the same method to obtain the solutions of the differential equations

(a) (b) (2) dx TIP X-P dt X-P ZIP || || xp -d dt Xp dt = (dr) (x - - ) |2x