A-Level CIE Further Maths 8.2 Second Order Differential Equations: Obtain the general solution of t

Obtain the general solution of the differential equation $$\frac{d^2x}{dt^2} - 6\frac{dx}{dt} + 25x = 19\sin 2t.$$ - CIE - A-Level Further Maths - Question 4 - 2014 - Paper 1

Question 4

$Obtain-the-general-solution-of-the-differential-equation--$$\frac{d^2x}{dt^2}---6\frac{dx}{dt}-+-25x-=-19\sin-2t.$$--CIE-A-Level Further Maths-Question 4-2014-Paper 1.png$

Obtain the general solution of the differential equation $$\frac{d^2x}{dt^2} - 6\frac{dx}{dt} + 25x = 19\sin 2t.$$

Worked Solution & Example Answer:Obtain the general solution of the differential equation $$\frac{d^2x}{dt^2} - 6\frac{dx}{dt} + 25x = 19\sin 2t.$$ - CIE - A-Level Further Maths - Question 4 - 2014 - Paper 1

Step 1

Find the Characteristic Equation

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Answer

Start with the homogeneous part of the differential equation:

$\frac{d^2x}{dt^2} - 6\frac{dx}{dt} + 25x = 0.$ The characteristic equation is given by:

$m^2 - 6m + 25 = 0.$ Using the quadratic formula, we find:

$m = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 25}}{2 \cdot 1} = \frac{6 \pm \sqrt{36 - 100}}{2} = 3 \pm 4i.$

Step 2

Find the Complementary Function (CF)

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Answer

The complementary function is:

$x_c = e^{3t}(A \cos 4t + B \sin 4t),$ where A and B are arbitrary constants.

Step 3

Find the Particular Integral (PI)

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Answer

To find the particular integral, we assume:

$x_p = -p \sin 2t + q \cos 2t.$ Differentiating, we find:

$\frac{dx_p}{dt} = -2p \cos 2t - 2q \sin 2t,$

$\frac{d^2x_p}{dt^2} = -4p \sin 2t + 4q \cos 2t.$ Substituting these into the differential equation leads to:

$-4p \sin 2t + 4q \cos 2t - 6(-2p \cos 2t - 2q \sin 2t) + 25(-p \sin 2t + q \cos 2t) = 19\sin 2t.$

Step 4

Simplify and Solve for p and q

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Answer

Combining terms gives the equation:

$-12p + 12q = 19,$

$-12p + 21q = 0.$ Solving this system yields:

$p = 7, \quad q = 4.$

Step 5

General Solution (GS)

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Answer

Now that we have p and q, the general solution of the differential equation is:

$x(t) = e^{3t}(A \cos 4t + B \sin 4t) + 7 \sin 2t + 4 \cos 2t.$

Obtain the general solution of the differential equation $$\frac{d^2x}{dt^2} - 6\frac{dx}{dt} + 25x = 19\sin 2t.$$ - CIE - A-Level Further Maths - Question 4 - 2014 - Paper 1

Worked Solution & Example Answer:Obtain the general solution of the differential equation $$\frac{d^2x}{dt^2} - 6\frac{dx}{dt} + 25x = 19\sin 2t.$$ - CIE - A-Level Further Maths - Question 4 - 2014 - Paper 1

Find the Characteristic Equation

Find the Complementary Function (CF)

Find the Particular Integral (PI)

Simplify and Solve for p and q

General Solution (GS)

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