2021-12-28

Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problem. Primes denote derivatives with respect to x.
$\frac{dy}{dx}={\left(64xy\right)}^{\frac{1}{3}}$

deginasiba

$\frac{dy}{{y}^{\frac{1}{3}}}={\left(64\right)}^{\frac{1}{3}}\left({x}^{\frac{1}{3}}\right)dx$
$\int \frac{dy}{{y}^{\frac{1}{3}}}=\int 4\left({x}^{\frac{1}{3}}\right)dx$
$\frac{3{y}^{\frac{2}{3}}}{2}=\frac{3\cdot 4{x}^{\frac{4}{3}}}{4}+C$
$\frac{3{y}^{\frac{2}{3}}}{2}=3{x}^{\frac{4}{3}}+C$
${y}^{\frac{2}{3}}=2{x}^{\frac{4}{3}}+\frac{2}{3}C$
$y={\left(2{x}^{\frac{4}{3}}+\frac{2}{3}C\right)}^{\frac{3}{2}}$
Here, C is the constant of integration,
Suppose $\frac{2}{3}$ is equal to another constant D
Rewrite the solution as
$y={\left(2{x}^{\frac{4}{3}}+D\right)}^{\frac{3}{2}}$ which is the general explicit solution the given differential equation.

temnimam2

$x-\left(\frac{7}{9}\right)=\frac{13}{9}$
$\frac{67}{100}=0$
$4\cdot {\left(\mathrm{ln}\left(x\right)\right)}^{3}$
$p=\frac{w}{f}$
$\frac{6.5}{12}=0$

karton

$\frac{dy}{dx}=\left(64xy{\right)}^{\frac{1}{3}}=4{x}^{\frac{1}{3}}{y}^{\frac{1}{3}}$
or, ${y}^{-\frac{1}{3}}dy=4{x}^{\frac{1}{3}}dx$
Integrating, we get,
$\int {y}^{-\frac{1}{3}}dy=4\int {x}^{\frac{1}{3}}dx$
or, $\frac{{y}^{-\frac{1}{3}}+1}{-\frac{1}{3}+1}=4\frac{{x}^{\frac{1}{3}}+1}{\frac{1}{3}+1}+c$
or, $\frac{{y}^{\frac{2}{3}}}{\frac{2}{3}}=4\frac{{x}^{\frac{4}{3}}}{\frac{4}{3}}+c$
or, $\frac{3}{2}{y}^{2/3}=3{x}^{4/3}+c$
which is the required general solution, where c be an arbitrary constant.
$\left(1+x\right)\frac{dy}{dx}=4y$
or, $\frac{dy}{y}=4\frac{dx}{x+1}$
Integrating, $\int \frac{dy}{y}=4\int \frac{dx}{x+1}$
or, $\mathrm{ln}|y|=4\mathrm{ln}|x+1|+\mathrm{ln}c$
or, $y=c\left(x+1{\right)}^{4}$
Required general solution:
$y=c\left(x+1{\right)}^{4}$ where c be an arbitrary constant.

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