Differential equations
Solve the following differential equations using methods for …
- separable
- first order linear
Tasks
1. \( \dfrac{du}{dt} = 35 + 7u + 5t + ut \).
\( \dfrac{du}{dt} = (5 + u)(7 + t) \)
2. \( y' + xy = x \)
\( \alpha(x) = e^{\frac{x^2}{2}} \)
or separable: \( \frac{dy}{dx} = (1-y)x \)
3. \( y' + \dfrac{1}{x+1}y = x^{-2} \)
\( \alpha(x) = x + 1 \)
4. \( t y' + 2y = t^2 - t + 1 \)
\( \alpha(x) = t^2 \)
5. \( y' = x e^{-\sin x} - y \cos x \)
\( \alpha(x) = e^{\sin x}\)
Answers (mixed up)
- \( y = e^{-\sin x} (\dfrac{x^2}{2} + C) \)
- \( y = 1 + C \cdot e^{\frac{-x^2}{2}} \)
- \( y = (\ln|x| - x^{-1} + C)(x+1)^{-1} \)
- \( y = \dfrac{t^2}{4} - \dfrac{t}{3} + \dfrac{1}{2} + \dfrac{C}{t^2} \)
- \( u = -5 + C e^{\frac{t^2}{2} + 7t} \)