Primitives et équations différentielles – exercices de maths Terminale

Exercice 1Vérifier

1. \(F(x) = (x - 1)e^{x}\) primitive de \(xe^{x}\) ?
2. \(G(x) = \dfrac{1}{2}(\ln x)^2\) primitive de \(\dfrac{\ln x}{x}\) ?

Exercice 2Usuelles

1. \(4x^3 - 6x + 1\)
2. \(\dfrac{3}{x^2}\)
3. \(2e^{x} - \dfrac{5}{x}\)
4. \(\dfrac{2}{\sqrt{x}}\)
5. \(3\cos x + \sin x\)

Exercice 3Composées

1. \(e^{-2x + 3}\)
2. \(3x^2(x^3 - 2)^5\)
3. \(\dfrac{4x}{2x^2 + 1}\)
4. \(\dfrac{x}{\sqrt{x^2 + 9}}\)
5. \(xe^{x^2}\)

Exercice 4Condition initiale

1. \(f(x) = 3x - 5\), \(F(2) = 1\) ?
2. \(g(x) = e^{3x}\), \(G(0) = 1\) ?
3. \(h(x) = \dfrac{1}{x}\), \(H(1) = 3\) ?

Exercice 5y' = ay

1. \(y' = -2y\)
2. \(y' = 0{,}3y\), \(y(0) = 5\)
3. \(3y' - y = 0\), \(y(3) = 2\)

Exercice 6y' = ay + b

1. \(y' = 3y - 6\)
2. \(y' = -2y + 4\), \(y(0) = 3\)
3. \(y' + 5y = 10\), \(y(0) = 0\)

Exercice 7Particulière

\(y' = 2y + e^{3x}\).

1. Solution \(ke^{3x}\) ?
2. Homogène ?
3. Générale, puis \(y(0) = 1\) ?

Exercice 8Refroidissement

\(T' = -0{,}08(T - 18)\), \(T(0) = 90\).

1. Solution ?
2. \(T(15)\) ?
3. \(T < 22\) quand ?

Exercice 9Décroissance

\(N' = -\lambda N\), \(N(0) = N_0\).

1. \(N(t)\) ?
2. Demi-vie en fonction de \(\lambda\) ?
3. Iode 131 : \(8\) jours. \(\lambda\) ?

Exercice 10Problème

\(P' = 0{,}04P - 3\), \(P(0) = 100\).

1. Solution ?
2. Limite, interprétation ?
3. Si \(P(0) = 50\) ?