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Solve the linear differential equation 2xy' + y = 2√x

User Jhrmnn
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1 Answer

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21 votes

Answer:

Explanation:

General form of the linear differential equation can be written as:


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

For this case, we can rewrite the equation as:


(dy)/(dx)+(1)/(2x)y=(√(x))/(x)

Here
P(x) =(1)/(2x); Q(x)=(√(x))/(x)

To find the solution (y(x)), we can use the integration factor method:


Fy(x)=\int Q(x)Fdx+C \rightarrow F=e^{\int P(x)dx

Then
F=e^{\int (1)/(2x)dx}=e^\right=√(|x|)

So, we can find:


y√(|x|)=\int (√(x)√(|x|))/(x)dx+C

Suppose that
x\in \double R, then
√(|x|)=√(x) , and we find:


y√(x)=x+C \rightarrow y(x)=√(x)+(C√(x))/(x)

To check our solution is right or not, put your y(x) back to the ODE:


y' = (1)/(2√(x))-\frac{C}{2\sqrt{x^(3)}}


2xy'=(x-C)/(√(x))


2xy'+y=(x-C)/(√(x))+√(x)+(C√(x))/(x)=2√(x)

(it means your solution is right)

User Lucas Grugru
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