How to solve this first order linear differential equation?
#xy'1/(x+1)y=x #
y(1) = 0
(According to our professor, I.F. = #e^(intf(x))# , and we should just leave the integral be for now if the integral cannot be solved by hand or conventional methods)
y(1) = 0
(According to our professor, I.F. =
# y = x/(x+1)(x + lnx 1) #
We have:
We can use an integrating factor when we have a First Order Linear nonhomogeneous Ordinary Differential Equation of the form;
So, we can put the equation in standard form:
Then the integrating factor is given by;
We can readly evaluate this integral if we perform a partial fraction decomposition of the integrand:
Then:
So we can write:
This is now separable, so by "separating the variables" we get:
Which is trivial to integrate to get the General Solution:
Leading to the Particular Solution:
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.
A first Order linear Differential Equation has the form of:
The integration factor is:
We can use partial fraction expansion to solve it:
Now, we multiply both sides of our ODE by this integration factor:
Then, we simplify and refine:
We now take the integral of both sides:
Now, we can apply the initial conditions:
Therefore,
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To solve a firstorder linear differential equation, follow these steps:

Write the equation in standard form: dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x.

Identify the integrating factor, μ(x), defined as e^(∫P(x)dx).

Multiply both sides of the equation by the integrating factor, μ(x).

Integrate both sides of the equation with respect to x.

Solve for y to find the general solution.

If an initial condition (boundary condition) is given, use it to find the particular solution by substituting the given values into the general solution.

If needed, simplify the solution further.
These steps provide a systematic approach to solve firstorder linear differential equations.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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