Quentin Johnson

2022-01-22

Solve the equation separable, linear, bernoulli, or homogenous
$1.\frac{dy}{dx}=\frac{{x}^{4}+4x{y}^{2}}{2{x}^{3}+{x}^{2}y+{y}^{3}}$
$2.\left({e}^{-y}\mathrm{cos}\left(x\right)\right){y}^{\prime }={x}^{4}+6{x}^{2}{y}^{3}$
$3.{y}^{\prime }=\frac{y+y{x}^{3}}{x+{x}^{2}}\mathrm{cos}\left(\frac{{x}^{2}}{{y}^{2}}\right)$

Hector Roberts

Expert

Separable equation:
A first order differential equation y’=f(x,y) is called a separable equation if the function f(x,y) can be factored into the product of two functions of x and y
$f\left(x,y\right)={f}_{1}\left(x\right){f}_{2}\left(y\right)$
where ${f}_{1}\left(x\right)$ and ${f}_{2}\left(y\right)$ are continuous functions.
Linear differential equation:
A first order differential equation is linear when it can be written as:
$\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$
Where P(x) and Q(x) are functions of x.
Bernoulli differential equation:
A first order Bernoulli differential equation can be written as: $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$
Where P(x) and Q(x) are functions of x.
Bernoulli differential equation:
A first order Bernoulli differential equation can be written as: $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right){y}^{n}$
Where P(x) and Q(x) are functions of x.
Homogeneous differential equation:
A first order homogeneous differential equation can be written as: $\frac{dy}{dx}=F\left(\frac{y}{x}\right)$
All three equations are written in the table :
\[\begin{array}{|c|c|}\hline Equation & Separable & Linear & Bernoilli & Homogeneous\ \hline \frac{dy}{dx}=\frac{x^4+4xy^2}{2x^3+x^2y+y^3} & no & no & no & no\\hline (e^{-y}\cos(x))y=x^4+6x^2y^3

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