Question

Solve the equation (D^2-4D+3) y=cos2x

Answer 1

Final Answer:

The differential equation $(D^2-4D+3) y=cos2x$ is solved to obtain the result $y=c_1e^{3x}+c_2e^{x}-\frac{8sin2x}{65}-\frac{cos2x}{65}$.

Given:

The differential equation $(D^2-4D+3) y=cos2x$ is provided.

To Find:

The differential equation $(D^2-4D+3) y=cos2x$ is to be solved.

Explanation:

The following points are important to solve the present problem.

• The differential equation is given in the form $f(D)y=F(x)$, where D denotes the derivatives of the dependent variable $y(x)$.
• The complete solution of the differential equation $f(D)y=F(x)$ is obtained by the sum of the complimentary function (C.F) and the particular integral (P.I) $y=C.F+P.I$.
• When there is the differential equation of the form $f(D)y=0$ , its solution is given by $y=C.F$.
• The C.F is obtained by the solution of the auxiliary equation (A.E) formed by the derivatives of the dependent variable $y(x)$ in the differential equation.

Step 1 of 4

As per the statement in the given problem, write and solve the following auxiliary equation (A.E).

$m^2-4m+3 =0\\m^2-3m-m+3=0\\m(m-3)-1(m-3)=0\\(m-3)(m-1)=0\\m=3,1$

Step 2 of 4

From the above solution to the A.E and the provided differential equation, its complimentary function (C.F) is as follows with two arbitrary constants $c_1, c_2$.

$C.F=c_1e^{3x}+c_2e^{x}$

Step 3 of 4

Again, from the provided differential equation, its particular integral (P.I) is the following.

$P.I\\=\frac{1}{f(D)} F(x)\\=\frac{1}{(D^2-4D+3)}cos2x\\=\frac{1}{(-4-4D+3)}cos2x\;\;\;\;[D^2=-2^2=-4]\\=\frac{1}{-(4D+1)}cos2x\\=-\frac{(4D-1)}{(4D)^2-1^2}cos2x\\=-\frac{(4D-1)}{65}cos2x\;\;\;\;[D^2=-2^2=-4]\\=-\frac{4D.cos2x}{65}-\frac{cos2x}{65}\\=-\frac{8sin2x}{65}-\frac{cos2x}{65}$

Step 4 of 4

Thus the complete solution of the provided differential equation is

$y=C.F+P.I\\y=c_1e^{3x}+c_2e^{x}-\frac{8sin2x}{65}-\frac{cos2x}{65}$

Therefore, the required solution of the differential equation $(D^2-4D+3) y=cos2x$ is $y=c_1e^{3x}+c_2e^{x}-\frac{8sin2x}{65}-\frac{cos2x}{65}$ with two arbitrary constants $c_1, c_2$.

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