Dividing vs Subtracting to Solve Equations w/1 Variable

[find_the_fun]

If you have the equation \(\displaystyle \frac{dx}{dt}=4(x^2+1)\) I sometimes get confused if i should should subtract \(\displaystyle 4(x^2+1)\) from both sides or multiply by it's reciprocal. If I subtract from both sides then I'd have 0 on the right side and that would give a different answer after integration but mathematically why would it be wrong?

 

[chisigma]

Re: Dividing vs subtracting to get all terms with one variable to one side of equation

If you have the equation \(\displaystyle \frac{dx}{dt}=4(x^2+1)\) I sometimes get confused if i should should subtract \(\displaystyle 4(x^2+1)\) from both sides or multiply by it's reciprocal. If I subtract from both sides then I'd have 0 on the right side and that would give a different answer after integration but mathematically why would it be wrong?

Since for all real x is $ 4\ (x^{2} + 1) > 0$ You can divide both sides by $ 4\ (x^{2} + 1)$ without danger and then integrate separately in x and y...

Kind regards

$\chi$ $\sigma$

 

[MarkFL]

Re: Dividing vs subtracting to get all terms with one variable to one side of equation

If you have the equation \(\displaystyle \frac{dx}{dt}=4(x^2+1)\) I sometimes get confused if i should should subtract \(\displaystyle 4(x^2+1)\) from both sides or multiply by it's reciprocal. If I subtract from both sides then I'd have 0 on the right side and that would give a different answer after integration but mathematically why would it be wrong?

Since this equation is separable, the best approach would be to divide so that you can separate variables, and as chisigma pointed out, you can do this without worrying about division by zero, and thus you are eliminating no solutions.