What is the order of the differential equation d3x dt3 t3 cos 2 dx dt Convert it into system of first order equations. Do not attempt to find a solution [2 marks]
The Correct Answer and Explanation is:
Correct Answer:
The given differential equation is:d3xdt3+t3cos(2)⋅dxdt=0\frac{d^3x}{dt^3} + t^3 \cos(2) \cdot \frac{dx}{dt} = 0dt3d3x+t3cos(2)⋅dtdx=0
Order of the differential equation:
The order of a differential equation is the highest derivative present. Here, the highest derivative is d3xdt3\frac{d^3x}{dt^3}dt3d3x, so:
Order = 3
To convert it into a system of first-order equations:
Let:
- x1=xx_1 = xx1=x
- x2=dxdt=dx1dtx_2 = \frac{dx}{dt} = \frac{dx_1}{dt}x2=dtdx=dtdx1
- x3=d2xdt2=dx2dtx_3 = \frac{d^2x}{dt^2} = \frac{dx_2}{dt}x3=dt2d2x=dtdx2
From the original equation:d3xdt3+t3cos(2)⋅dxdt=0\frac{d^3x}{dt^3} + t^3 \cos(2) \cdot \frac{dx}{dt} = 0dt3d3x+t3cos(2)⋅dtdx=0
We substitute:d3xdt3=dx3dt=−t3cos(2)⋅x2\frac{d^3x}{dt^3} = \frac{dx_3}{dt} = -t^3 \cos(2) \cdot x_2dt3d3x=dtdx3=−t3cos(2)⋅x2
So the system becomes:{dx1dt=x2dx2dt=x3dx3dt=−t3cos(2)⋅x2\begin{cases} \frac{dx_1}{dt} = x_2 \\ \frac{dx_2}{dt} = x_3 \\ \frac{dx_3}{dt} = -t^3 \cos(2) \cdot x_2 \end{cases}⎩⎨⎧dtdx1=x2dtdx2=x3dtdx3=−t3cos(2)⋅x2
Explanation
This is a third-order ordinary differential equation (ODE) because the highest derivative involved is d3xdt3\frac{d^3x}{dt^3}dt3d3x. Identifying the order of an ODE is important as it tells us how many initial conditions are required to determine a unique solution. In this case, three initial conditions would be necessary: the values of xxx, dxdt\frac{dx}{dt}dtdx, and d2xdt2\frac{d^2x}{dt^2}dt2d2x at some initial time.
Transforming a higher-order ODE into a system of first-order equations is a standard technique used to make the equation more suitable for numerical or theoretical analysis. By defining new variables for each derivative up to the second order, we reduce the third-order equation to three coupled first-order equations. This approach is widely used in engineering, physics, and applied mathematics, especially for simulation and modeling using software that typically solves only first-order systems.
In this conversion, we introduced three variables: x1x_1x1, x2x_2x2, and x3x_3x3, representing the function and its first two derivatives. This allowed us to express the original third derivative as a first-order derivative of x3x_3x3, and relate it to x2x_2x2 and ttt using the original equation. The resulting system captures the full dynamics of the original third-order equation.
