Separable differential equations problems

**Obstacle1** · August 13th 2007, 03:36 AM

Two problems I'm stuck on.( sorry these aren't separable, they are first order linear, i wrote the wrong thing)

1: t*(dy/dt) + 2y = t^2 - t + 1

2: t*(dy/dt) + (t+1)*y = t

Any help finding the general solutions would be much appreciated.
I already got the integrating factors as 1: exp(2lnt) and 2: exp(t+lnt)

**galactus** · August 13th 2007, 05:08 AM

For #1:

$ty'+2y=t^{2}-t+1$

Divide by t: $y'+\frac{2}{t}y=t-1+\frac{1}{t}$

The integrating factor is 2/t.

$e^{\int{\frac{2}{t}}dt}=t^{2}$

$\frac{d}{dt}[t^{2}y]=t^{2}(t-1+\frac{1}{t})=t^{3}-t^{2}+t$

Integrate:

$t^{2}y=\frac{t^{4}}{4}-\frac{t^{3}}{3}+\frac{t^{2}}{2}+C$

Solve for y:

$y=\frac{t^{2}}{4}-\frac{t}{3}+\frac{1}{2}+Ct^{-2}$

**Krizalid** · August 13th 2007, 05:42 AM

Originally Posted by Obstacle1

Two problems I'm stuck on.( sorry these aren't separable, they are first order linear, i wrote the wrong thing)

1: t*(dy/dt) + 2y = t^2 - t + 1

2: t*(dy/dt) + (t+1)*y = t

Any help finding the general solutions would be much appreciated.
I already got the integrating factors as 1: exp(2lnt) and 2: exp(t+lnt)

When you have a linear ODE like $y'+P(x)y=Q(x)$ the integrating factor is given by $\color{blue}\mu(x)=\exp\int P(x)\,dx$

Then you amplify the equation by $\mu$ to set the left side into a product's function derivative, finally all you have to do is integrate.

**Krizalid** · August 13th 2007, 10:50 AM

Originally Posted by Obstacle1

I already got the integrating factors as 2: exp(t+lnt)

$\exp(t+\ln t)=\exp(\ln e^t+\ln t)=t\cdot e^t$

That's all.

Thread: Separable differential equations problems

LinkBack

Thread Tools

Display

Separable differential equations problems

Similar Math Help Forum Discussions

Separable Differential Equations Problems

Separable Differential Equations

Separable Differential Equation problems

Separable Differential Equations-Please help!

First Order Differential Equations: Separable Equations and Applications

Search Tags