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Use the method of integrating factors to solve the following initial value problems.y'+xy=x, y(0)=1y'+3x2y= sin(x)e-x^2, y(0)=1

Question

Use the method of integrating factors to solve the following initial value problems.

  • y'+xy=x, y(0)=1
  • y'+3x2y= sin(x)e-x^2, y(0)=1
check_circleAnswer
Step 1

The given equation are 

yxyxy01
y3x2ynx)ey(0)1
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yxyxy01 y3x2ynx)ey(0)1

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Step 2

Solve the initial value problem

yxyy0)1
Note that, the given equation first order linear differential equation which is in the form
of y' p(x)y(x), where p (x) x and q(x)
Step 3:
Compute the integrating factor as follows
That is, the integrating factor ux) = e2.
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yxyy0)1 Note that, the given equation first order linear differential equation which is in the form of y' p(x)y(x), where p (x) x and q(x) Step 3: Compute the integrating factor as follows That is, the integrating factor ux) = e2.

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Step 3

Multiply the integrating factor on bot...

e? (у'+ ху) — х:е?
x2
х
e? у+ хуе ? %3Dх-е?
2
d
e у 3 х:е
dx
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e? (у'+ ху) — х:е? x2 х e? у+ хуе ? %3Dх-е? 2 d e у 3 х:е dx

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