how to Find equilibrium points and plot phase portrait of the following system of differential equations

Posted inODEs
Question : Find equilibrium points and plot phase portrait of the following system of differential equations x’=x(8-4x-y) y’=y(6-x-2y) Solution: Given system of differential equations x'=x(8-4x-y)                           (1) y'=y(6-x-2y)                           (2) We shall…

how to solve ode y”=3y’-10y y(0)=1 y'(0)=-1

Posted inAdvanced Math, ODEs
Question : solve ode y''=3y'-10y y(0)=1 y'(0)=-1 Solution : Given homogeneous linear differential equation \[y''=3y'-10y\] \[ y(0)=1,\ y'(0)=-1\] \[y''-3y'+10y=0 \] (1) So it’s auxiliary equation is \[\lambda^2-3\lambda+10=0\] \[\lambda^2-5\lambda+2\lambda+10=0\] \[\lambda\left(\lambda-5\right)+2\left(\lambda-5\right)=0\] \[\left(\lambda+2\right)\left(\lambda-5\right)=0\]…

how to solve differential equation y”+2y’-8y=0

Posted inAdvanced Math, ODEs
Question: Solve differential equation y''+2y'-8y=0 , y(0)=1, y'(1)=1 Given homogeneous linear differential equation \[y''=-2y'+8y\] \[\ y(0)=1,\ y'(0)=1 \] \[y''+2y'-8y=0 \] (1) So it’s auxiliary equation is \[ \lambda^2+2\lambda-8=0\] \[\lambda^2+4\lambda-2\lambda-8=0\] \[\lambda\left(\lambda+4\right)-2\left(\lambda+4\right)=0\]…

which ode does y=sin(log(x)) satisfy?

Posted inAbstract Algebra, Advanced Math, Mathematics, Multivariable Calculus, ODEs
Question : Which ode does y=sin(log(x)) satisfy? Solution : Given function \[y=sin\left(log\left(x\right)\right)\] differentiating with respect to x \[\frac{dy}{dx}=\cos{\left(\log{\left(x\right)}\right)}\frac{1}{x}\] \[x\ \frac{dy}{dx}=\cos{\left(\log{\left(x\right)}\right)}\] Again differentiating with respect to x \[\frac{dy}{dx}+x\frac{d^2y}{dx^2}=-sin\left(log\left(x\right)\right)\frac{1}{x}=\frac{y}{x}\] \[x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+y=0\] is the…