y'=6x^2/(2y+cosy))

y'=6x^2/(2y+cosy))

Question
Laplace transform
asked 2021-03-11
\(\displaystyle{y}'={6}\frac{{x}^{{2}}}{{{2}{y}+{\cos{{y}}}}}{)}\)

Answers (1)

2021-03-12
This is a separable equation, which can be written as \(\displaystyle{\left({2}{y}+{\cos{{y}}}\right)}{\left.{d}{y}\right.}={6}{x}^{{2}}{\left.{d}{x}\right.}\)
Integrate: \(\displaystyle∫{\left({2}{y}+{\cos{{y}}}\right)}{\left.{d}{y}\right.}=∫{6}{x}^{{2}}{\left.{d}{x}\right.}\)
LHS: \(\displaystyle∫{\left({2}{y}+{\cos{{y}}}\right)}{\left.{d}{y}\right.}={y}^{{2}}+{\sin{{y}}}+{C}{1},\) where C1 is some constant.
RHS: \(\displaystyle∫{6}{x}^{{2}}{\left.{d}{x}\right.}={2}{x}^{{3}}+{C}{2}\) where C2 is some constant.
So, from (1) it follows that \(\displaystyle{y}^{{2}}+{\sin{{y}}}={2}{x}^{{3}}+{C}\) where we defined C as C2-C1
0

Relevant Questions

asked 2021-03-07

Solve differential equation \(\frac{\cos^2y}{4x+2}dy= \frac{(\cos y+\sin y)^2}{\sqrt{x^2+x+3}}dx\)

asked 2020-12-24
Use Laplace transform to solve the folowing initial value problem \(y"+2y'+2y=0\)
\(y(0)=2\)
\(y'(0)=-1\)
asked 2021-02-19
Use Laplace transform to solve the following initial-value problem
\(y"+2y'+y=0\)
\(y(0)=1, y'(0)=1\)
a) \displaystyle{e}^{{-{t}}}+{t}{e}^{{-{t}}}\)
b) \displaystyle{e}^{t}+{2}{t}{e}^{t}\)
c) \displaystyle{e}^{{-{t}}}+{2}{t}{e}^{t}\)
d) \displaystyle{e}^{{-{t}}}+{2}{t}{e}^{{-{t}}}\)
e) \displaystyle{2}{e}^{{-{t}}}+{2}{t}{e}^{{-{t}}}\)
f) Non of the above
asked 2021-01-30
Use Laplace transform to find the solution of the IVP
\(2y'+y=0 , y(0)=-3\)
a) \(f{{\left({t}\right)}}={3}{e}^{{-{2}{t}}}\)
b)\(f{{\left({t}\right)}}={3}{e}^{{\frac{t}{{2}}}}\)
c)\(f{{\left({t}\right)}}={6}{e}^{{{2}{t}}}
d) \(f{{\left({t}\right)}}={3}{e}^{{-\frac{t}{{2}}}}\)
asked 2021-01-27
Please solve the 2nd order differential equation by (PLEASE FOLLOW GIVEN METHOD) LAPLACE TRANSFORMATION
ALSO, USE PARTIAL FRACTION WHEN YOU ARRIVE
\(L(y) = \left[\frac{w}{(s^2 + a^2)(s^2+w^2)}\right]*b\)
Problem 2 Solve the differential equation
\(\frac{d^2y}{dt^2}+a^2y=b \sin(\omega t)\) where \(y(0)=0\)
and \(y'(0)=0\)
asked 2021-03-07
use the Laplace transform to solve the initial value problem.
\(y"-3y'+2y=\begin{cases}0&0\leq t<1\\1&1\leq t<2\\ -1&t\geq2\end{cases}\)
\(y(0)=-3\)
\(y'(0)=1\)
asked 2021-01-31
Find the solutions for the given linear systems of differential equations using Laplace Transforms.
\(w'-2y'+3w=0: y"+w=2\sin x\)
\(y(0)=w(0)=2 , y’(0)=-1\)
asked 2021-01-25
Use the Laplace transform to solve the given system of differential equations.
\(\frac{(d^2x)}{(dt^2)}+\frac{(d^2y)}{(dt^2)}=\frac{t}{2}\)
\(\frac{(d^2x)}{(dt^2)}-\frac{(d^2y)}{(dt^2)}=4t\)
\(x(0) = 5, x'(0) = 0,\)
\(y(0) = 0, y'(0) = 0\)
asked 2020-11-09
Solve the third-order initial value problem below using the method of Laplace transforms
\(y'''-2y"-21y'-18y=-18\)
\(y(0)=2\)
\(y'(0)=7\)
\(y"(0)=95\)
asked 2021-01-22
Use Laplace transform to solve the folowing initial value problem \(y"+2y'+y=4e^{-t} y(0)=2 y'(0)=-1\)
...