What is the particular solution of the differential equation? : #y'+4xy=e^(-2x^2)# with #y(0)=-4.3#

Answer 1

# y = (x-4.3)e^(-2x^2) #

We have:

# y' +4xy = e^(-2x^2) # ..... [A]

We can use an integrating factor when we have a First Order Linear non-homogeneous Ordinary Differential Equation of the form;

# dy/dx + P(x)y=Q(x) #

The given equation is already in standard form, so the integrating factor is given by;

# I = e^(int P(x) dx) # # \ \ = exp(int \ 4x \ dx) # # \ \ = exp( 2x^2 ) # # \ \ = e^(2x^2) #
And if we multiply the DE [A] by this Integrating Factor, #I#, we will have a perfect product differential;
# 2e^(2x^2)y' +8xe^(2x^2)y = 2e^(2x^2)e^(-2x^2) #
# :. d/dx (2e^(2x^2)y ) = 2 #

Which we can now "seperate the variables" to get:

# 2e^(2x^2)y = int \ 2 \ dx #

Which is trivial to integrate giving the General Solution:

# 2e^(2x^2)y = 2x + C #

Applying the initial condition we get:

# 2e^(0)(-4.3) = 0 + C => C = -8.6#

Giving the Particular Solution:

# 2e^(2x^2)y = 2x -8.6 # # :. e^(2x^2)y = x -4.3 # # :. e^(2x^2)e^(-2x^2)y = (x-4.3)e^(-2x^2)# # :. y = (x-4.3)e^(-2x^2) #
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Answer 2

The particular solution of the given differential equation is:

[ y(x) = e^{-2x^2} - 4.3 - \frac{1}{2}x^2 e^{-2x^2} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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