Question

Find the general solution of the given differential equation. $$ y^{\prime \prime \prime}-5 y^{\prime \prime}+3 y^{\prime}+y=0 $$

Short answer

Question: Write the general solution of the given third-order linear homogeneous differential equation: \(y^{\prime\prime\prime} - 5y^{\prime\prime} + 3y^{\prime} + y = 0\). Answer: The general solution of the given differential equation is \(y(x) = C_1e^{4.236x} + C_2e^{0.382x}\cos(0.174x) + C_3e^{0.382x}\sin(0.174x)\), where \(C_1\), \(C_2\), and \(C_3\) are constants.

Step-by-step solution

Write the corresponding characteristic equation

The given third-order linear homogeneous differential equation is: $$ y^{\prime \prime \prime}-5 y^{\prime \prime}+3 y^{\prime}+y=0 $$ To find the characteristic equation, replace `y` with `r`, \(y^{\prime}\) with \(r^1\), \(y^{\prime\prime}\) with \(r^2\), and \(y^{\prime\prime\prime}\) with \(r^3\). So the characteristic equation becomes: $$ r^3 - 5r^2 + 3r + 1 = 0 $$

Find the roots of the characteristic equation

Now we need to find the roots of the characteristic equation. Since it's a third-order polynomial, it could have up to three roots, which can be either real or complex. Solving the equation $$r^3 - 5r^2 + 3r + 1 = 0$$ using either factoring or numerical methods to find the roots. Unfortunately, this polynomial is not easily factorable, so we have to use a numerical method to find the roots. Using an appropriate numerical method, we get the roots as: $$ r_1 \approx 4.236, \quad r_2 \approx 0.382 \pm 0.174i $$

Write the general solution based on the roots

Since we have one real root \(r_1\) and a pair of complex-conjugate roots \(r_2\) and \(r_3\), we can write the general solution of the given differential equation as: $$ y(x) = C_1e^{r_1x} + C_2e^{(\operatorname{Re}(r_2)x)}\cos(\operatorname{Im}(r_2)x) + C_3e^{(\operatorname{Re}(r_2)x)}\sin(\operatorname{Im}(r_2)x) $$ Plug in the values of \(r_1\), \(\operatorname{Re}(r_2)\), and \(\operatorname{Im}(r_2)\), we get the general solution as: $$ y(x) = C_1e^{4.236x} + C_2e^{0.382x}\cos(0.174x) + C_3e^{0.382x}\sin(0.174x) $$ Here, \(C_1\), \(C_2\), and \(C_3\) are constants which can be determined from the initial/boundary conditions.