Chapter 9
For each of the differential equations given below, indicate its order and degree (if defined). MATHEMA TICS334 (i) 22 2 5 6 logd y dy x y xdxdx + − = (ii) 3 2 4 7 sindy dy y xdx dx − + = (iii) 4 3 4 3sin 0d y d y dx dx − =
For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation. (i) xy = a ex + b e- x + x2 : 2/2 2 2 2 0d y dyx xy x dxdx + − + − = (ii) y = ex (a cos x + b sin x) : 2 2 2 0d y dy ydxdx − + = (iii) y = x sin 3x : 2 9 6cos3 0d y y x dx + − = (iv) x2 = 2y2 log y : 2 2( ) 0 dyx y xy dx+ − =
Prove that x2 - y2 = c (x2 + y2)2 is the general solution of differential equation (x3 - 3 x y2) dx = ( y3 - 3 x2y) dy, where c is a parameter .
Find the general solution of the differential equation 2/2 1 0 dy y dx x −+ = − .
Show that the general solution of the differential equation 2/2 1 0
| dy | y | y |
|---|---|---|
| dx | x | x |
Find the equation of the curve passing through the point 0, 4 π whose differential equation is sin x cos y dx + cos x sin y dy = 0.
Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0.
Solve the differential equation 2 ( 0) x x y yy e dx x e y dy y = + ≠ .
Find a particular solution of the differential equation (x - y) (dx + dy) = dx - dy, given that y = -1, when x = 0. (Hint: put x - y = t)
Solve the differential equation 1( 0) xe y dx xdyx x − − = ≠ .
Find a particular solution of the differential equation cotdy y xdx + = 4x cosec x (x ≠ 0), given that y = 0 when 2x π= .
Find a particular solution of the differential equation (x + 1) dy dx = 2 e- y - 1, given that y = 0 when x = 0.
The general solution of the differential equation 0y dx x dy y − = is (A) xy = C (B) x = C y2 (C) y = Cx (D) y = C x2
The general solution of a differential equation of the type 1 1P Qdx xdy + = is (A) ( ) / 1 1P P / 1Q C dy dy y e e dy∫ ∫ = +∫ (B) ( ) / 1 1P P