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

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! ^_^

### Re: Differential Eqn. Elimination of arbitrary constant and...

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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