Toronto Math Forum
MAT2442013F => MAT244 MathTests => Quiz 1 => Topic started by: Qian Li on October 05, 2013, 03:40:39 PM

3 from section 2.6, i.e.
$$
(3x^22xy+2)+(6y^2x^2+3)y'=0
$$
We already know it is exact, partial derivative of $(3x^22xy+2)$ with respect to $y$ is $2x$, while partial derivative of
$(6y^2x^2+3)$ with respect to $x$ is also $2x$.
So just assume the original function is $F(x,y)$. $P(x,y)=3x^22xy+2$ is obtained from $F(x,y)$ by taking patial derivative with respect to $x$, and $Q(x,y)=(6y^2x^2+3)$ with respect to $y$.
Now integrate $P(x,y)$ with respect to $x$, and we can get $F(x,y)=x^3x^2y+2x+h(y)$. Take partial derivative with respect to $y$, and we get $Q(x,y)= x^2+h'(y)$. ERROR!!!
If compare this $Q(x,y)$ with the $6y^2x^2+3$, we get $h'(y)=6y^2+3$, which means $h(y)=2y^3+3y+C$, where $C$ is any arbitrary constant.
In conclusion, $F(x,y)=x^3x^2y+2x+2y^3+3y+C$, or we can write it as $x^3x^2y+2x+2y^3+3y=C$.
Do not hijack topicsthis was about Q1, P1so I split it

Quan Li, next time correcting solution write a response after error.