Marenonigt

2022-01-22

Important

Solve the bernoullis

Solve the bernoullis

Jeremy Merritt

Beginner2022-01-22Added 31 answers

Make the given differential equation into the standard form of first order ordinary differential equation. Now determine the integrating factor.

Substitute it in the solution form. Simplify the equation in proper form.

First rewrite the equation and convert it into first order ordinary differential equation.

$\frac{dy}{dx}({x}^{2}{y}^{3}+xy)=1$

$\frac{dy}{dx}=\frac{1}{({x}^{2}{y}^{3}+xy)}$

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

$\frac{dx}{dy}-xy={x}^{2}{y}^{3}$

$\frac{1}{{x}^{2}}\frac{dx}{dy}-\frac{y}{x}={y}^{3}$

Now let$\frac{1}{x}=u$ then on differentiating this equation on both the sides with respect to y, $-\frac{1}{{x}^{2}}\frac{dx}{dy}=\frac{du}{dy}$ .

Substituting it in the equation, it will be$\frac{du}{dx}+uy=-{y}^{3}$ .

Substitute it in the solution form. Simplify the equation in proper form.

First rewrite the equation and convert it into first order ordinary differential equation.

Now let

Substituting it in the equation, it will be

braodagxj

Beginner2022-01-23Added 38 answers

Now this is an equation of first order differential equation $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$ thus determine the integrating factor (IF) determine as $IF={e}^{\int P\left(x\right)dx}$ where the solution is $y{e}^{\int P\left(x\right)dx}=-\int Q\left(x\right){e}^{\int P\left(x\right)dx}dx.$

$IF={e}^{\int ydx}={e}^{\frac{{y}^{2}}{2}}$

Substitute it in the solution form and simplify.

$u{e}^{\frac{{y}^{2}}{2}}=-\int {y}^{2}{e}^{\frac{{y}^{2}}{2}}dy$

$\frac{{e}^{\frac{{y}^{2}}{2}}}{x}=-2(\frac{{y}^{2}}{2}-1){e}^{\frac{{y}^{2}}{2}}+c$

$\frac{1}{x}=2-{y}^{2}+c{e}^{-\frac{{y}^{2}}{2}}$

Hence the solution is$\frac{1}{x}=2-{y}^{2}+c{e}^{-\frac{{y}^{2}}{2}}$

Substitute it in the solution form and simplify.

Hence the solution is