Solve the following first-order linear differential equations? $y\prime -4{x}^{3}y=8{x}^{3}$

zapri4j
2022-09-23
Answered

Solve the following first-order linear differential equations? $y\prime -4{x}^{3}y=8{x}^{3}$

You can still ask an expert for help

Absexabbelpjl

Answered 2022-09-24
Author has **8** answers

The ODE is

$y\prime -4{x}^{3}y=8{x}^{3}$

$\frac{dy}{dx}=8{x}^{3}+4{x}^{3}y=4{x}^{3}(2+y)$

$\frac{dy}{y+2}=4{x}^{3}dx$

$\int \frac{dy}{y+2}=\int 4{x}^{3}dx$

$\mathrm{ln}(y+2)={x}^{4}+{C}_{1}$

$y+2={e}^{{x}^{4}+{C}_{1}}={e}^{{C}_{1}}{e}^{{x}^{4}}=C{e}^{{x}^{4}}$

$y=C{e}^{{x}^{4}}-2$

$y\prime -4{x}^{3}y=8{x}^{3}$

$\frac{dy}{dx}=8{x}^{3}+4{x}^{3}y=4{x}^{3}(2+y)$

$\frac{dy}{y+2}=4{x}^{3}dx$

$\int \frac{dy}{y+2}=\int 4{x}^{3}dx$

$\mathrm{ln}(y+2)={x}^{4}+{C}_{1}$

$y+2={e}^{{x}^{4}+{C}_{1}}={e}^{{C}_{1}}{e}^{{x}^{4}}=C{e}^{{x}^{4}}$

$y=C{e}^{{x}^{4}}-2$

asked 2021-01-15

Show that the first order differential equation $(x+1){y}^{\prime}-3y=(x+1{)}^{5}$ is of the linear type.

Hence solve for y given that y = 1.5 when x = 0

Hence solve for y given that y = 1.5 when x = 0

asked 2022-06-21

Consider $x{y}^{\u2033}+2{y}^{\prime}+xy=0$. Its solutions are $\frac{\mathrm{cos}x}{x},\phantom{\rule{thinmathspace}{0ex}}\frac{\mathrm{sin}x}{x}$.

Neither of those solutions (as far as I could find) can be the solutions of a first order linear homogeneous differential equation. However

$\frac{{e}^{\pm ix}}{x}$

Can be the solution of a first order linear homogeneous DE (${y}^{\prime}+(x\mp i)y=0$)

Can I always find a first order homogeneous linear DE whose solution also solves a second order homogeneous linear DE?

For example can I find a first order homogeneous linear DE whose solution is a particular linear combination of ${J}_{1}$ and ${Y}_{1}$?

(unrelated: also I'd like to know if there is a way of solving $x{y}^{\u2033}+2{y}^{\prime}+xy=0$ without noticing that it is a spherical bessel function or using laplace transform.)

Neither of those solutions (as far as I could find) can be the solutions of a first order linear homogeneous differential equation. However

$\frac{{e}^{\pm ix}}{x}$

Can be the solution of a first order linear homogeneous DE (${y}^{\prime}+(x\mp i)y=0$)

Can I always find a first order homogeneous linear DE whose solution also solves a second order homogeneous linear DE?

For example can I find a first order homogeneous linear DE whose solution is a particular linear combination of ${J}_{1}$ and ${Y}_{1}$?

(unrelated: also I'd like to know if there is a way of solving $x{y}^{\u2033}+2{y}^{\prime}+xy=0$ without noticing that it is a spherical bessel function or using laplace transform.)

asked 2022-05-29

How can I write the following equation as a first-order vector differential equation?

$\begin{array}{r}m\frac{{\mathrm{d}}^{2}x}{{\mathrm{d}}^{2}t}+2\gamma m\frac{\mathrm{d}x}{\mathrm{d}t}+kx=0.\end{array}$

$\begin{array}{r}m\frac{{\mathrm{d}}^{2}x}{{\mathrm{d}}^{2}t}+2\gamma m\frac{\mathrm{d}x}{\mathrm{d}t}+kx=0.\end{array}$

asked 2022-11-16

Solve the Differential Equation $\frac{dy}{dx}=2xy-x$?

asked 2021-03-05

Solve differential equation
$dy/dx-12{x}^{3}y={x}^{3}$

asked 2022-05-27

If I have a differential equation ${y}^{\prime}(t)=Ay(t)$ where A is a constant square matrix that is not diagonalizable(although it is surely possible to calculate the eigenvalues) and no initial condition is given. And now I am interested in the fundamental matrix. Is there a general method to determine this matrix? I do not want to use the exponential function and the Jordan normal form, as this is quite exhausting. Maybe there is also an ansatz possible as it is for the special case, where this differential equation is equivalent to an n-th order ode. I saw a method where they calculated the eigenvalues of the matrix and depending on the multiplicity n of this eigenvalue they used an exponential term(with the eigenvalue) and in each component an n-th order polynomial as a possible ansatz. Though they only did this, when they were interested in a initial value problem, so with an initial condition and not for a general solution.

I was asked to deliver an example: so ${y}^{\prime}(t)=\left(\begin{array}{cc}3& -4\\ 1& -1\end{array}\right)y(t)$ If somebody can construct a fundamental matrix for this system, than this should be sufficient

I was asked to deliver an example: so ${y}^{\prime}(t)=\left(\begin{array}{cc}3& -4\\ 1& -1\end{array}\right)y(t)$ If somebody can construct a fundamental matrix for this system, than this should be sufficient

asked 2022-01-19

Classify the equation as separable, linear, both, or neither.

$y\frac{dy}{dx}=2x+y$