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Initial value problem

d2y/dx2-3xdy/dx+2x=0, x0=6 d2y/dx2=-5
(i) Solve the problem using the standard form of the general solution, based on the characteristic
equation.
(ii) Solve the problem as a system of rst order ODEs, by letting y1 = X and y2 = y01

User Vinesh
by
3.1k points

1 Answer

16 votes
16 votes

Since you explicitly mention "characteristic equation", I assume you mean the differential equation is


(d^2y)/(dx^2) - 3(dy)/(dx) + 2y = 0

because the phrase is often used in the context of linear ODEs with constant coefficients.

The characteristic equation is then


r^2 - 3r + 2 = (r - 2) (r - 1) = 0

with roots
r=2 and
r=1. Then the characteristic solution is


\boxed{y = C_1 e^(2x) + C_2 e^x}

By substituting
z = y' and
z' = y'', the ODE transforms to


(dz)/(dx) - 3z + 2y = 0

so we have the system of ODEs


\begin{cases} z' = 3z - 2y \\ y' = z \end{cases}

In matrix form,


\frac d{dx} \begin{bmatrix} z \\ y \end{bmatrix} = \begin{bmatrix} 3 & -2 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} z \\ y \end{bmatrix}

Compute the eigenvalues of the coefficient matrix.


\begin{vmatrix} 3 - \lambda & -2 \\ 1 & -\lambda \end{vmatrix} = \lambda^2 - 3\lambda + 2 = (\lambda - 2) (\lambda - 1) = 0 \\\\ \implies \lambda = 2 \text{ or } \lambda = 1

Compute the corresponding eigenvectors.


\lambda = 2 \implies \begin{bmatrix} 1 & -2 \\ 1 & -2 \end{bmatrix} \vec v = \vec 0 \implies \vec v = \begin{bmatrix}2 \\ 1\end{bmatrix}


\lambda = 1 \implies \begin{bmatrix}2 & -2 \\ 1 & -1 \end{bmatrix} \vec v = \vec 0 \implies \vec v = \begin{bmatrix}1 \\ 1 \end{bmatrix}

Then the characteristic solution to the system is


\begin{bmatrix} z \\ y \end{bmatrix} = C_1 e^(2x) \begin{bmatrix} 2 \\ 1 \end{bmatrix} + C_2 e^x \begin{bmatrix}1 \\ 1 \end{bmatrix}

but we only want the second component,


\boxed{y = C_1 e^(2x) + C_2 e^x}

You also mention "initial value problem" but it's unclear what the initial values are. It looks like you might have meant
y(0)=6 and
y'(0) = -5, in which case


y(0) = C_1 + C_2 = 6


y'(0) = 2C_1 + C_2 = -5

By elimination,


(C_1 + C_2) - (2C_1 + C_2) = 6 - (-5) \implies -C_1 = 11 \\\\ \implies C_1 = -11 \text{ and } C_2 = 17

and so the particular solution is


y = -11 e^(2x) + 17 e^x

You did write d2y/dx2=-5, though, so you also could have meant
y''(0) = -5, n which case


y(0) = C_1 + C_2 = 6


y''(0) = 4C_1 + C_2 = -5


\!\!\implies (C_1 + C_2) - (4C_1 + C_2) = 6 - (-5) \implies -3C_1 = 11\\\\ \implies C_1 = -\frac{11}3 \text{ and } C_2 = \frac{29}3

Take your pick.

User Mateen Ulhaq
by
2.6k points