GATE ME 2018 Set 2 — Question 46
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Engineering Mathematics → Differential Equations → Higher-Order Linear ODEs
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Question
Given the ordinary differential equationwith and , the value of is _________ (correct to two decimal places).
Correct answer
1.45 to 1.48
Solution
The given ordinary differential equation is a second-order linear homogeneous equation with constant coefficients:Step 1: Find the characteristic equation.
Substituting into the differential equation gives the characteristic equation:The roots are and .Step 2: Write the general solution.
The general solution is a linear combination of the basis solutions:Step 3: Apply initial conditions to find constants.
Using :Using :
First, find the derivative of the general solution:At :Substitute :Then, .Step 4: Calculate the value of .
The specific solution is .
At :Using and :Rounding to two decimal places, we get .
Substituting into the differential equation gives the characteristic equation:The roots are and .Step 2: Write the general solution.
The general solution is a linear combination of the basis solutions:Step 3: Apply initial conditions to find constants.
Using :Using :
First, find the derivative of the general solution:At :Substitute :Then, .Step 4: Calculate the value of .
The specific solution is .
At :Using and :Rounding to two decimal places, we get .
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