Francisca Rodden

2021-12-29

Test whether the differential equations are exact and solve those that are $3{x}^{2}{y}^{2}dx+\left(2x{y}^{3}+4{y}^{3}\right)dy=0$

amarantha41

Given differential equation
$3{x}^{2}{y}^{2}dx+\left(2x{y}^{3}+4{y}^{3}\right)dy=0$ (1)
Comparing (1) with $Mdx+Ndy=0$ we get
$M=3{x}^{2}{y}^{2},N=2{x}^{3}x+4{y}^{3}$ Now $\frac{\partial M}{\partial y}=\frac{\partial }{\partial y}\left(3{x}^{2}{y}^{2}\right)=3{x}^{2}\cdot \frac{\partial }{\partial y}\left({y}^{2}\right)=3{x}^{2}×2y=6{x}^{2}y$
$\frac{\partial N}{\partial x}=\frac{\partial }{\partial x}\left(2{x}^{3}y+4{y}^{3}\right)=2y\frac{\partial }{\partial x}\left({x}^{3}\right)+\frac{\partial }{\partial x}\left(4{y}^{3}\right)$
$=2y×3{x}^{2}+0$
$=6{x}^{2}y$
Since $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}=6{x}^{2}y$
So the differential equation (1) is exact
Since the given differential equation exact so the solution
$\int Mdx\left(y=\text{constant}\right)+\int N\left(\text{the term not contained}x\right)dy=c$
$⇒\int 3{x}^{2}{y}^{2}dx+\int 4{y}^{3}dy=C$
$y=\text{constant}$
$⇒3{y}^{2}\int {x}^{2}dx+4\frac{{y}^{4}}{4}=c$
$⇒3{y}^{2}×\frac{{x}^{3}}{3}+\frac{4{y}^{4}}{4}=c$
$⇒{y}^{2}{x}^{3}+{y}^{4}=c$
Hence the solution is ${y}^{2}{x}^{3}+{y}^{4}=c$

Annie Gonzalez

Simplifying
$3{x}^{2}{y}^{2}\cdot dx+\left(2{x}^{3}y+4{y}^{3}\right)\cdot dy=0$
Multiply ${x}^{2}{y}^{2}\cdot dx$
$3{dx}^{3}{y}^{2}+\left(2{x}^{3}y+4{y}^{3}\right)\cdot dy=0$
Reorder the terms for easier multiplication:
$3{dx}^{3}{y}^{2}+dy\left(2{x}^{3}y+4{y}^{3}\right)=0$
$3{dx}^{3}{y}^{2}+\left(2{x}^{3}y\cdot dy+4{y}^{3}\cdot dy\right)=0$
$3{dx}^{3}{y}^{2}+\left(2{dx}^{3}{y}^{2}+4{dy}^{4}\right)=0$
Combine like terms: $3{dx}^{3}{y}^{2}+2{dx}^{3}{y}^{2}=5{dx}^{3}{y}^{2}$
$5{dx}^{3}{y}^{2}+4{dy}^{4}=0$
Solving
$5{dx}^{3}{y}^{2}+4{dy}^{4}=0$
Solving for variable 'd'.
Move all terms containing d to the left, all other terms to the right.
Factor out the Greatest Common Factor (GCF), '${dy}^{2}$'.
${dy}^{2}\left(5{x}^{3}+4{y}^{2}\right)=0$

karton