Question

Solve the following differential equation by using linear equations. dx/dt = 1- t + x - tx

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asked 2021-05-07
Solve the following differential equation by using linear equations.
\(dx/dt = 1- t + x - tx\)

Answers (1)

2021-05-08
Step 1
The given differential equation is,
\(\frac{dx}{dt}=1-t+x-tx\)
For solving the given differential equation using linear equation, first we separate the terms.
\(\frac{dx}{dt}=1-t+x-tx\)
\(\frac{dx}{dt}=(1+x)-t(1+x)\)
\(\frac{dx}{dt}=(1+x)(1-t)\)
\(\frac{dx}{(1+x)}=(1-t)dt\)
Step 2
Applying integration, on both the sides, we get
\(\int \frac{1}{(1+x)}dx=\int (1-t)dt\)
\(\ln(1+x)=t-\frac{t^{2}}{2}+C\)
\(1+x=e^{(t-\frac{t^{2}}{2}+C)}\)
\(x=e^{(t-\frac{t^{2}}{2}+C)}-1\)
Therefore, the general solution of the given differential equation is \(x=e^{(t-\frac{t^{2}}{2}+C)}-1\)
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