Differential Eqn. Elimination of arbitrary constant and finding the general solution

hello guys, please help me with my assignment. thank you so much.

1. eliminate the arbituary constant on this eqn. y=C1e^2x+C2e^x+C3

2. evaluate and find the general solution of this eqn.
a. x sqrt1+y^2 dx - y sqrt1+x^2 dy=0
b. 2ycosxdx+3sinxdy = 0

thank you so much. god bless! ^_^

Jhun Vert's picture

Problem 1
$y =C_1 e^{2x} + C_2 e^x + C_3$   ←   eq (1)

There are 3 arbitrary constants C1, C2, and C3, hence, we can differentiate eq (1) up to three times.
$y' = 2C_1 e^{2x} + C_2 e^x$   ←   eq (2)

$y'' = 4C_1 e^{2x} + C_2 e^x$   ←   eq (3)

$y''' = 8C_1 e^{2x} + C_2 e^x$   ←   eq (4)
 

eq (3) - eq(2)
$y'' - y' = 2C_1 e^{2x}$   ←   eq (5)

eq (4) - eq(3)
$y''' - y'' = 4C_1 e^{2x}$   ←   eq (6)

eq (6) - 2×eq (5)
$(y''' - y'') - 2(y'' - y') = 0$

$y''' - y'' - 2y'' + 2y' = 0$

$y''' - 3y'' + 2y' = 0$       answer

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