Q6 E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 3-8, determine whether the given function is a solution to the given differential equation.

$\mathbf{x}\mathbf{=}\mathbf{cos}\mathbf{ }\mathbf{2}\mathbf{t}$, $\frac{\mathbf{dx}}{\mathbf{dt}}\mathbf{+}\mathbf{tx}\mathbf{=}\mathbf{sin}\mathbf{ }\mathbf{2}\mathbf{t}$

Step-by-Step Solution

Answer

The given function is not a solution to the given differential equation.

Step 1: Differentiating the given equation w.r.t. (with respect to) t.

Firstly, we will differentiate $x=\cos 2t$ with respect to t,

$\frac{\mathrm{dx}}{\mathrm{dt}}=-2 \sin 2\mathrm{t}$

Step 2: Simplification.

Putting the values from step 1 in the L.H.S. (Left-hand side) of the given differential equation,

$\frac{\mathrm{dx}}{\mathrm{dt}}+\mathrm{tx}=-2 \sin 2\mathrm{t}+\mathrm{t} \cos 2\mathrm{t}$

which is not the same as the R.HS. (Right-hand side) of the given differential equation.

Hence, $\mathbf{x}\mathbf{=}\mathbf{cos}\mathbf{ }\mathbf{2}\mathbf{t}$ is not a solution to the differential equation $\frac{\mathbf{dx}}{\mathbf{dt}}\mathbf{+}\mathbf{tx}\mathbf{=}\mathbf{sin}\mathbf{ }\mathbf{2}\mathbf{t}$.