Consider the initial value problem below. The value of y at x = In 2, (rounded off to 3 decimal places) is dy/dx = 2x - y, y(0) = 1 (Important - Enter only the numerical value in the answer)Correct answer is between '0.8774,0.8952'. Can you explain this answer?

Question

Answer

Concept:

The standard form of a first-order linear differential equation is,

Where P and Q are the functions of x.

Integrating factor, IF = e^∫Pdx

Now, the solution for the above differential equation is,

y(IF) = ∫IF.Qdx

Calculation:

By comparing the above differential equation with the standard differential equation,

P = 1, Q = 2x

Integrating factor, IF = e^∫Pdx = e^∫1dx = e^x

Now, the solution is

y(e^x) = ∫e^x(2x)dx

ye^x = 2(xe^x− e^x) + C

⇒ y = 2(x − 1) + Ce^−x

y(0) = 1

⇒ 1 = 2 (0 – 1) + C

⇒ C = 3

Now, the solution becomes

y = 2x – 2 + 3e^-x

At x = ln 2,

y = 2 ln 2 – 2 + 3 (0.5) = 0.8862

Answer

Explanation:
Understanding the Initial Value Problem:
- The initial value problem provided is dy/dx = 2x - y, with the initial condition y(0) = 1.
- This problem involves finding the value of y at x = ln(2) using the given differential equation and initial condition.
Solving the Initial Value Problem:
- To solve the initial value problem, we need to find the solution to the differential equation dy/dx = 2x - y with the initial condition y(0) = 1.
- By solving the differential equation, we can find the function y(x) that satisfies the given equation.
- Substituting the initial condition y(0) = 1 into the solution will give us the value of y at x = ln(2).
Finding the Exact Value of y at x = ln(2):
- Solving the differential equation and applying the initial condition y(0) = 1 will give us the exact value of y at x = ln(2).
- The calculated value, rounded off to 3 decimal places, falls between 0.8774 and 0.8952.

Conclusion:
- The value of y at x = ln(2), obtained by solving the initial value problem, lies between 0.8774 and 0.8952.
- This range represents the approximate numerical value of y at x = ln(2) based on the given differential equation and initial condition.

Similar Civil Engineering (CE) Doubts

Consider the initial value problem given below. the value of y at x=ln2 (rounded off to one decimal plac e) is____dy/dx = 2x − y, y(0)=1

Consider the differential equation dy/dx = 4(x + 2) - yFor the initial condition y = 3 at x = 1, the value of y at x = 1.4 obtained using Euler’s method with a step-size of 0.2 is ________ . (round off to one decimal plac e) Correct answer is between '6.3,6.5'. Can you explain this answer?

Giventhe value of y at x = 2 is ________ (round off to nearest integer) (Important -Enter only the numerical value in the answer) Correct answer is between '162,170'. Can you explain this answer?

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