Expert answers to "Find the general solution of the differential equation..."

Humberto Winters

Answered question

2022-02-12

Find the general solution of the differential equation y′′ − y′ = 2te2t. Hint: Substitute x = y′ and solve it for x first

Emilie Mcpherson

Beginner2022-02-13Added 12 answers

y^{'} = x, y^{″} = x^{'}

So,

\frac{α x}{α t} - x = 2 t e^{2 t}

P (t) = - 1, Q (t) = 2 t e^{2 t}

= e^{- \int d t} = e^{- t}

x \times e^{- t} = 2 \int t e^{2 t} \cdot e^{- t} d t = 2 \int t e^{t} d t

x e^{- t} = 2 [t e^{t} - e^{t}] + c

x = 2 [t e^{2 t} - e^{2 t}] + c e^{t}

\frac{d y}{d t} = 2 t e^{2 t} - 2 e^{2 t} + c e^{t}

Now again integrate

y = [\frac{2 t e^{2 t}}{2} - \frac{e^{2 t}}{2}] - \frac{2 e^{2 t}}{2} + c e^{t} + c^{'}

y = \frac{2 t e^{2 t}}{2} - \frac{3}{2} e^{2 t} + c e^{t} + c^{'}

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?