If xy+6e^{y} = 6e, find the value of y" at the point where x = 0.

Question
Differential equations
asked 2020-11-12
If \(\displaystyle{x}{y}+{6}{e}^{{{y}}}={6}{e}\), find the value of y" at the point where x = 0.

Answers (1)

2020-11-13
Here the Differential equations is given by xy+6e^y=6e with y(0)=1y(0)=1. Taking differentiate both sides with respect to xx we get (d/dx(xy+6e^y))=(d/dx)6e^y (d/dx(xy)+6(d/dx)e^y=0 y+xy'+(6e^y)y'=0 From here putting y(0)=1y(0)=1 we get y′(0)=−1/6e. Agin differentiate both sides above expressions with respect to x we get (d/dx)(y+xy'+(6e^y)y')=0 y'+y'+xy''+(6e^y)y'^2+(6e^y)y''=0 Now putting x=0x=0 and y(0)=1y(0)=1, y′(0)=−1/6e we get -(1/3e)-6e(1/36e^2)+6ey''=0 -> 6ey''=1/3e+1/6e -> 6ey''=1/2e -> y''=1/12e^2
0

Relevant Questions

asked 2020-11-08
A particle moves along the curve \(\displaystyle{x}={2}{t}^{{2}}{y}={t}^{{2}}-{4}{t}\) and z=3t-5 where t is the time.find the components of the velocity at t=1 in the direction i-3j+2k
asked 2021-01-02
Solve this equation pls \(\displaystyle{y}'+{x}{y}={e}^{{x}}\)
\(\displaystyle{y}{\left({0}\right)}={1}\)
asked 2021-01-02
Find dw/dt using the appropriate Chain Rule. Evaluate \(\frac{dw}{dt}\) at the given value of t. Function: \(w=x\sin y,\ x=e^t,\ y=\pi-t\) Value: t = 0
asked 2020-10-26
Find the differential dy for the given values of x and dx. \(y=\frac{e^x}{10},x=0,dx=0.1\)
asked 2021-02-05
Solve the differential equations
(1) \(\displaystyle{x}{y}'-{2}{y}={x}^{{3}}{e}^{{x}}\)
(2) \(\displaystyle{\left({2}{y}{\left.{d}{x}\right.}+{\left.{d}{y}\right.}\right)}{e}^{{2}}{x}={0}\)
asked 2021-01-04
Solve the initial value problem.
\(\displaystyle{4}{x}^{{2}}{y}{''}+{17}{y}={0},{y}{\left({1}\right)}=-{1},{y}'{\left({1}\right)}=-\frac{{1}}{{2}}\)
asked 2021-02-11
Solve the Bernoulli Differential equations. \(\displaystyle{x}{y}′-{y}={x}{y}^{{5}}\)
asked 2021-02-21
The coefficient matrix for a system of linear differential equations of the form \(\displaystyle{y}^{{{1}}}={A}_{{{y}}}\) has the given eigenvalues and eigenspace bases. Find the general solution for the system.
\(\displaystyle{\left[\lambda_{{{1}}}=-{1}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}{0}{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},\lambda_{{{2}}}={3}{i}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}-{i}{1}+{i}{7}{i}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},\lambda_{{3}}=-{3}{i}\Rightarrow\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}+{i}{1}-{i}-{7}{i}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}\right]}\)
asked 2020-11-12
\(F(x) = (lnx)^{\cos}x,\) find f'(x) and the equation to the tangent line at (e 1)
asked 2020-12-16
\(\displaystyle{\left({2}{y}+{x}{y}\right)}{\left.{d}{x}\right.}+{2}{x}{\left.{d}{y}\right.}={0}\)
...