Question

Найти общее и частное решение y''-8y'=0

Answer 1

$y''-8y'=0$

Составим и решим характеристическое уравнение:

$\lambda^2-8\lambda=0$

$\lambda(\lambda-8)=0$

$\lambda_1=0$

$\lambda-8=0\Rightarrow \lambda_2=8$

Запишем общее решение:

$y=C_1e^{\lambda_1 x}+C_2e^{\lambda_2 x}$

$y=C_1e^{0x}+C_2e^{8x}$

$\boxed{y=C_1+C_2e^{8x}}$

Запишем частное решение. Пусть $C_1=1;\ C_2=2$:

$\boxed{y=1+2e^{8x}}$

Answer 2

General and Particular Solutions of y'' - 8y' = 0

The given differential equation is y'' - 8y' = 0. To find the general and particular solutions, let's proceed step by step.

Step 1: Find the Characteristic Equation To find the characteristic equation, we assume that the solution is in the form of y = e^(rx), where r is a constant. Taking the derivatives of y, we have y' = re^(rx) and y'' = r^2e^(rx). Substituting these into the differential equation, we get:

r^2e^(rx) - 8re^(rx) = 0

Factoring out e^(rx), we have:

e^(rx)(r^2 - 8r) = 0

Since e^(rx) is never zero, we can divide both sides by e^(rx):

r^2 - 8r = 0

This is the characteristic equation.

Step 2: Solve the Characteristic Equation To solve the characteristic equation r^2 - 8r = 0, we factor out an r:

r(r - 8) = 0

This equation has two solutions: r = 0 and r = 8.

Step 3: Find the General Solution The general solution of the differential equation is given by:

y = C1e^(r1x) + C2e^(r2x)

Substituting the values of r1 = 0 and r2 = 8, we have:

y = C1e^(0x) + C2e^(8x)

Simplifying further, we get:

y = C1 + C2e^(8x)

This is the general solution of the given differential equation.

Step 4: Find the Particular Solution To find the particular solution, we need additional information such as initial conditions or boundary conditions. Without any additional information, we cannot determine a unique particular solution.

In summary, the general solution of the differential equation y'' - 8y' = 0 is y = C1 + C2e^(8x), where C1 and C2 are arbitrary constants. However, without specific initial or boundary conditions, we cannot determine a unique particular solution.

I hope this helps! Let me know if you have any further questions.