Holooly Plus Logo
Holooly Plus Logo

Question 12.1.6: Write the following initial value problem as a first-order s...

Write the following initial value problem as a first-order system:

t^{3} x^{\prime \prime \prime}+4 t^{2} x^{\prime \prime}-8 t x^{\prime}+8 x=0, \quad x(2)=3, \quad x^{\prime}(2)=-6, \quad x^{\prime \prime}(2)=14.

The Blue Check Mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

Defining x_{1}=x, x_{2}=x^{\prime}, x_{3}=x^{\prime \prime}, we obtain the system

\begin{aligned}&x_{1}{ }^{\prime}=x_{2}, \\&x_{2}{ }^{\prime}=x_{3}, \\&x_{3}{ }^{\prime}=\frac{-8}{t^{3}} x_{1}+\frac{8}{t^{2}} x_{2}-\frac{4}{t} x_{3},\end{aligned}

with the initial condition x_{1}(2)=3, x_{2}(2)=-6, x_{3}(2)=14.

Related Answered Questions

Solve the initial value problem x” + x = 8 sin t, x(0) = 3, x'(0) = 1.

Verified Answer:
Using the substitution x_{1}=x, x_{2}=x^{\p...

Find the unique solution to the system x’ = [x1 x2]’ = [4 2 3 3][x1 x2] + [e^t e^2t] = Ax + f(t), x(0) = [1 2].

Verified Answer:
A fundamental matrix solution for the homogeneous ...

(A Model of Species Cooperation-Symbiosis) Consider the symbiotic model governed by the system X1′(t) = -1/2 x1(t) + x2(t), x2′(t) = 1/4 x1(t) – 1/2 x2(t). In this model the population of each species increases proportionally to the population of the other and decreases proportionally to its own

Verified Answer:
Here A=\left(\begin{array}{rr}-\frac{1}{2} ...

(A Predator-Prey Model) Consider the predator-prey model governed by the system X1′(t) = X1(t) + X2(t), X2′(t) = -X1(t) + X2(t). If the initial populations are x1(0) = x2(0) = 1000, determine the populations of the two species for t > 0.

Verified Answer:
Here A=\left(\begin{array}{rr}1 & 1 \\ ...

(A Competitive Model) Consider the system x1′(t) = 3×1(t) – x2(t), x2′(t) = -2×1(t) + 2×2(t). Here, an increase in the population of one species causes a decline in the growth rate of another. Suppose that the initial populations are x1(0) = 90 and x2(0) = 150. Find the populations of both species

Verified Answer:
We have A=\left(\begin{array}{rr}3 & -1...

Find the general solution of the system x’ = 4x + y, y’ = -8x + 8y.

Verified Answer:
Here D=\left|\begin{array}{cc}4-\lambda &am...

Find the general solution of the system x’ = -4x – y, y’ = x – 2y.

Verified Answer:
Equation (3) is \begin{aligned}D &=\lef...

Find the general solution of the system x’ = -x + 6y, y’ = x – 2y.

Verified Answer:
Here a_{11}=-1, a_{12}=6, a_{21}=1, a_{22}=...

Solve the following system: x’ = 4x – y + t^2 y’ = x + 2y + 3t

Verified Answer:
Proceeding as before, we obtain \begin{alig...

Solve the system x’ = x + y, x(0) = 1, y’ = -3x – y, y(0) = 0.

Verified Answer:
Differentiating the first equation and substitutin...