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I want to solve this differential equation:

$$ C \cdot y(t)\frac{d}{dt} = x(t) - y(t) $$

$x(t)$ and $y(t)$ are two ordinary functions of t, C is a constant - all in in $R$

I am trying to solve it towards $y(t)$. The solution I am looking for looks something like this:

$$ y(t) = e^{\int{x(t)dt}} + C $$

So $\int{x(t)dt}$ can stay - but how will the rest look like and could you show me the individual steps and name the method how to solve it?

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    This comes very close to what I'm trying to do but my equation is a little bit simpler: http://de.wikipedia.org/wiki/Variation_der_Konstanten2011-06-24
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    What do you differentiate in your equation? The differentiation operator is written before the function, not after it. http://www.stewartcalculus.com/data/CALCULUS%20Concepts%20and%20Contexts/upfiles/3c3-LinearDiffEqns_Stu.pdf is a nice document you should take a look at.2011-06-24

1 Answers 1

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If you mean an equation $Cy'(t) = x(t)-y(t)$ then the solution is following.

If $C=0$ then $y = x$. If $C\neq 0$ then $y' + \frac 1 Cy = \frac{x}{C}$. You can solve it by the method "Variation of constant" or "Lagrange method".

  1. Solve the homogeneous equation: $$ y'+\frac{1}{C}y = 0. $$ Separate variables: $$ \frac{dy}{y} = -\frac 1 C dt. $$ By integration we obtain $$ y=K\exp\{-t/C\} $$ where $K$ is some constant.

  2. In Lagrange method you suppose that $K(t)$ is a function rather than a constant and substitute $y=K(t)\exp\{-t/C\}$ in the original equation $y'+\frac 1 C y = \frac x C$ to find $K(t).$ After substituition you obtain: $$ K'\exp\{-t/C\} = \frac{x}{C}, $$ so $$ K' = \frac{1}{C}x(t)\exp\{t/C\} $$ and $$ K = K_1+\frac{1}{C}\int x(t)\exp\{t/C\}dt. $$

Now, $$ y(t) = \left(K_1+\frac{1}{C}\int x(t)\exp\{t/C\}dt\right)\exp\{-t/C\} $$ where $K_1$ is some constant.

By the way, $\exp$ means exponent in this notation, $\exp\{a\} =\mathrm e^a$.

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    thank you! In the first equation of (2.): Where did the term "$+\frac{1}{C}y$" go?2011-06-24
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    More insights found here: http://www.voofie.com/content/6/introduction-to-differential-equation-and-solving-linear-differential-equations-using-operator-metho/#First_order_Linear_Differential_Equations2011-06-25