znacimavjo

2022-04-23

Solution of the differential equation $,f\left(x\right){f}^{\prime}\left(x\right)+{f}^{\prime}\left(x\right)=g\left(x\right)$ ?

How do we solve a differential equation of the form

$f\left(x\right){f}^{\prime}\left(x\right)+{f}^{\prime}\left(x\right)=g\left(x\right)?$

The coefficient f(x) of f'(x) post a difficulty that integrating factors doesn't work.

How do we solve a differential equation of the form

The coefficient f(x) of f'(x) post a difficulty that integrating factors doesn't work.

potomakavkl

Beginner2022-04-24Added 11 answers

Step 1

$g\left(x\right)=f\left(x\right){f}^{\prime}\left(x\right)+{f}^{\prime}\left(x\right)={(\frac{{f}^{2}\left(x\right)}{2}+f\left(x\right))}^{\prime}$

and hence, if$G\left(x\right)=\int g\left(x\right),dx$ , then

Step 2

$\frac{{f}^{2}\left(x\right)}{2}+f\left(x\right)=G\left(x\right)+c$

and thus$bi{g(f\left(x\right)+1big)}^{2}={f}^{2}\left(x\right)+2f\left(x\right)+1=2G\left(x\right)+2c+1$

Therefore$f\left(x\right)=\pm \sqrt{2G\left(x\right)+\stackrel{~}{c}}-1$

and hence, if

Step 2

and thus

Therefore

What is the derivative of the work function?

How to use implicit differentiation to find $\frac{dy}{dx}$ given $3{x}^{2}+3{y}^{2}=2$?

How to differentiate $y=\mathrm{log}{x}^{2}$?

The solution of a differential equation y′′+3y′+2y=0 is of the form

A) ${c}_{1}{e}^{x}+{c}_{2}{e}^{2x}$

B) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{3x}$

C) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{-2x}$

D) ${c}_{1}{e}^{-2x}+{c}_{2}{2}^{-x}$How to find instantaneous velocity from a position vs. time graph?

How to implicitly differentiate $\sqrt{xy}=x-2y$?

What is 2xy differentiated implicitly?

How to find the sum of the infinite geometric series given $1+\frac{2}{3}+\frac{4}{9}+...$?

Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?

A. 4.2

B. 4.4

C. 4.7

D. 5.1What is the derivative of $\frac{x+1}{y}$?

How to find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?

How to find the volume of a cone using an integral?

What is the surface area of the solid created by revolving $f\left(x\right)={e}^{2-x},x\in [1,2]$ around the x axis?

How to differentiate ${x}^{\frac{2}{3}}+{y}^{\frac{2}{3}}=4$?

The differential coefficient of $\mathrm{sec}\left({\mathrm{tan}}^{-1}\left(x\right)\right)$.