Analysis Beispiele

$\cos (x) \frac{d y}{d x} + y = \sin (x)$

Schritt 1

Schreibe die Differentialgleichung als $\frac{d y}{d x} + M (x) y = Q (x)$ um.

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Schritt 1.1

Teile jeden Ausdruck in $\cos (x) \frac{d y}{d x} + y = \sin (x)$ durch $\cos (x)$ .

$\frac{\cos (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos (x)} = \frac{\sin (x)}{\cos (x)}$

Schritt 1.2

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 1.2.1

Kürze den gemeinsamen Faktor.

$\frac{\cos (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos (x)} = \frac{\sin (x)}{\cos (x)}$

Schritt 1.2.2

Dividiere $\frac{d y}{d x}$ durch $1$ .

$\frac{d y}{d x} + \frac{y}{\cos (x)} = \frac{\sin (x)}{\cos (x)}$

Schritt 1.3

Faktorisiere $y$ aus $\frac{y}{\cos (x)}$ heraus.

$\frac{d y}{d x} + y \frac{1}{\cos (x)} = \frac{\sin (x)}{\cos (x)}$

Schritt 1.4

Stelle $y$ und $\frac{1}{\cos (x)}$ um.

$\frac{d y}{d x} + \frac{1}{\cos (x)} y = \frac{\sin (x)}{\cos (x)}$

Schritt 2

Der Integrationsfaktor ist definiert durch die Formel $e^{\int P (x) d x}$ , wobei $P (x) = \frac{1}{\cos (x)}$ gilt.

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Schritt 2.1

Stelle das Integral auf.

$e^{\int \frac{1}{\cos (x)} d x}$

Schritt 2.2

Integriere $\frac{1}{\cos (x)}$ .

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Schritt 2.2.1

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$e^{\int \sec (x) d x}$

Schritt 2.2.2

Das Integral von $\sec (x)$ nach $x$ ist $\ln (| \sec (x) + \tan (x) |)$ .

$e^{\ln (| \sec (x) + \tan (x) |) + C}$

Schritt 2.3

Entferne die Konstante der Integration.

$e^{\ln (\sec (x) + \tan (x))}$

Schritt 2.4

Exponentialfunktion und Logarithmusfunktion sind zueinander inverse Funktionen.

$\sec (x) + \tan (x)$

Schritt 3

Multipliziere jeden Ausdruck mit $\sec (x) + \tan (x)$ .

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Schritt 3.1

Multipliziere jeden Ausdruck mit $\sec (x) + \tan (x)$ .

$(\sec (x) + \tan (x)) \frac{d y}{d x} + (\sec (x) + \tan (x)) (\frac{1}{\cos (x)} y) = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2

Vereinfache jeden Term.

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Schritt 3.2.1

Vereinfache jeden Term.

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Schritt 3.2.1.1

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

$(\frac{1}{\cos (x)} + \tan (x)) \frac{d y}{d x} + (\sec (x) + \tan (x)) (\frac{1}{\cos (x)} y) = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.1.2

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$(\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) \frac{d y}{d x} + (\sec (x) + \tan (x)) (\frac{1}{\cos (x)} y) = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.2

Wende das Distributivgesetz an.

$\frac{1}{\cos (x)} \frac{d y}{d x} + \frac{\sin (x)}{\cos (x)} \frac{d y}{d x} + (\sec (x) + \tan (x)) (\frac{1}{\cos (x)} y) = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.3

Kombiniere $\frac{1}{\cos (x)}$ und $\frac{d y}{d x}$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x)}{\cos (x)} \frac{d y}{d x} + (\sec (x) + \tan (x)) (\frac{1}{\cos (x)} y) = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.4

Kombiniere $\frac{\sin (x)}{\cos (x)}$ und $\frac{d y}{d x}$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + (\sec (x) + \tan (x)) (\frac{1}{\cos (x)} y) = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.5

Vereinfache jeden Term.

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Schritt 3.2.5.1

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + (\frac{1}{\cos (x)} + \tan (x)) (\frac{1}{\cos (x)} y) = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.5.2

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) (\frac{1}{\cos (x)} y) = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.6

Kombiniere $\frac{1}{\cos (x)}$ und $y$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) \frac{y}{\cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.7

Wende das Distributivgesetz an.

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{1}{\cos (x)} \cdot \frac{y}{\cos (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{y}{\cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.8

Multipliziere $\frac{1}{\cos (x)} \cdot \frac{y}{\cos (x)}$ .

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Schritt 3.2.8.1

Mutltipliziere $\frac{1}{\cos (x)}$ mit $\frac{y}{\cos (x)}$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos (x) \cos (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{y}{\cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.8.2

Potenziere $\cos (x)$ mit $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{1} (x) \cos (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{y}{\cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.8.3

Potenziere $\cos (x)$ mit $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{1} (x) \cos^{1} (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{y}{\cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.8.4

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{{\cos (x)}^{1 + 1}} + \frac{\sin (x)}{\cos (x)} \cdot \frac{y}{\cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.8.5

Addiere $1$ und $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{y}{\cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.9

Multipliziere $\frac{\sin (x)}{\cos (x)} \cdot \frac{y}{\cos (x)}$ .

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Schritt 3.2.9.1

Mutltipliziere $\frac{\sin (x)}{\cos (x)}$ mit $\frac{y}{\cos (x)}$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos (x) \cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.9.2

Potenziere $\cos (x)$ mit $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{1} (x) \cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.9.3

Potenziere $\cos (x)$ mit $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{1} (x) \cos^{1} (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.9.4

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{{\cos (x)}^{1 + 1}} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.2.9.5

Addiere $1$ und $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.3

Vereinfache jeden Term.

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Schritt 3.3.1

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = (\frac{1}{\cos (x)} + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Schritt 3.3.2

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) \frac{\sin (x)}{\cos (x)}$

Schritt 3.4

Wende das Distributivgesetz an.

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Schritt 3.5

Multipliziere $\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Schritt 3.5.1

Mutltipliziere $\frac{1}{\cos (x)}$ mit $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos (x) \cos (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Schritt 3.5.2

Potenziere $\cos (x)$ mit $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{1} (x) \cos (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Schritt 3.5.3

Potenziere $\cos (x)$ mit $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{1} (x) \cos^{1} (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Schritt 3.5.4

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{{\cos (x)}^{1 + 1}} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Schritt 3.5.5

Addiere $1$ und $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Schritt 3.6

Multipliziere $\frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Schritt 3.6.1

Mutltipliziere $\frac{\sin (x)}{\cos (x)}$ mit $\frac{\sin (x)}{\cos (x)}$ .

Schritt 3.6.2

Potenziere $\sin (x)$ mit $1$ .

Schritt 3.6.3

Potenziere $\sin (x)$ mit $1$ .

Schritt 3.6.4

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

Schritt 3.6.5

Addiere $1$ und $1$ .

Schritt 3.6.6

Potenziere $\cos (x)$ mit $1$ .

Schritt 3.6.7

Potenziere $\cos (x)$ mit $1$ .

Schritt 3.6.8

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

Schritt 3.6.9

Addiere $1$ und $1$ .

Schritt 3.7

Vereinfache jeden Term.

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Schritt 3.7.1

Separiere Brüche.

$\frac{\frac{d y}{d x}}{1} \cdot \frac{1}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.7.2

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$\frac{\frac{d y}{d x}}{1} \sec (x) + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.7.3

Dividiere $\frac{d y}{d x}$ durch $1$ .

$\frac{d y}{d x} \sec (x) + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.7.4

Separiere Brüche.

$\frac{d y}{d x} \sec (x) + \frac{\frac{d y}{d x}}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.7.5

Wandle von $\frac{\sin (x)}{\cos (x)}$ nach $\tan (x)$ um.

$\frac{d y}{d x} \sec (x) + \frac{\frac{d y}{d x}}{1} \tan (x) + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.7.6

Dividiere $\frac{d y}{d x}$ durch $1$ .

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.7.7

Multipliziere mit $1$ .

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + \frac{y}{\cos^{2} (x) \cdot 1} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.7.8

Separiere Brüche.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + \frac{y}{1} \cdot \frac{1}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.7.9

Wandle von $\frac{1}{\cos^{2} (x)}$ nach $\sec^{2} (x)$ um.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + \frac{y}{1} \sec^{2} (x) + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.7.10

Dividiere $y$ durch $1$ .

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.7.11

Faktorisiere $\cos (x)$ aus $\cos^{2} (x)$ heraus.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + \frac{\sin (x) y}{\cos (x) \cos (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.7.12

Separiere Brüche.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + \frac{y}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.7.13

Wandle von $\frac{\sin (x)}{\cos (x)}$ nach $\tan (x)$ um.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + \frac{y}{\cos (x)} \tan (x) = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.7.14

Separiere Brüche.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + \frac{y}{1} \cdot \frac{1}{\cos (x)} \tan (x) = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.7.15

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + \frac{y}{1} \sec (x) \tan (x) = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.7.16

Dividiere $y$ durch $1$ .

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + y \sec (x) \tan (x) = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.8

Vereinfache jeden Term.

Tippen, um mehr Schritte zu sehen ...

Schritt 3.8.1

Faktorisiere $\cos (x)$ aus $\cos^{2} (x)$ heraus.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + y \sec (x) \tan (x) = \frac{\sin (x)}{\cos (x) \cos (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.8.2

Separiere Brüche.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + y \sec (x) \tan (x) = \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.8.3

Wandle von $\frac{\sin (x)}{\cos (x)}$ nach $\tan (x)$ um.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + y \sec (x) \tan (x) = \frac{1}{\cos (x)} \tan (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.8.4

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + y \sec (x) \tan (x) = \sec (x) \tan (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Schritt 3.8.5

Wandle von $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ nach $\tan^{2} (x)$ um.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + y \sec (x) \tan (x) = \sec (x) \tan (x) + \tan^{2} (x)$

Schritt 3.9

Stelle die Faktoren in $\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + y \sec (x) \tan (x) = \sec (x) \tan (x) + \tan^{2} (x)$ um.

$\sec (x) \frac{d y}{d x} + \tan (x) \frac{d y}{d x} + y \sec^{2} (x) + y \sec (x) \tan (x) = \sec (x) \tan (x) + \tan^{2} (x)$

Schritt 4

Schreibe die linke Seite als ein Ergebnis der Produktdifferenzierung.

$\frac{d}{d x} [(\sec (x) + \tan (x)) y] = \sec (x) \tan (x) + \tan^{2} (x)$

Schritt 5

Integriere auf beiden Seiten.

$\int \frac{d}{d x} [(\sec (x) + \tan (x)) y] d x = \int \sec (x) \tan (x) + \tan^{2} (x) d x$

Schritt 6

Integriere die linke Seite.

$(\sec (x) + \tan (x)) y = \int \sec (x) \tan (x) + \tan^{2} (x) d x$

Schritt 7

Integriere die rechte Seite.

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Schritt 7.1

Zerlege das einzelne Integral in mehrere Integrale.

$(\sec (x) + \tan (x)) y = \int \sec (x) \tan (x) d x + \int \tan^{2} (x) d x$

Schritt 7.2

Da die Ableitung von $\sec (x)$ gleich $\sec (x) \tan (x)$ ist, ist das Integral von $\sec (x) \tan (x)$ gleich $\sec (x)$ .

$(\sec (x) + \tan (x)) y = \sec (x) + C + \int \tan^{2} (x) d x$

Schritt 7.3

Schreibe $\tan^{2} (x)$ in $- 1 + \sec^{2} (x)$ um unter Verwendung des trigonometrischen Pythagoras.

$(\sec (x) + \tan (x)) y = \sec (x) + C + \int - 1 + \sec^{2} (x) d x$

Schritt 7.4

Zerlege das einzelne Integral in mehrere Integrale.

$(\sec (x) + \tan (x)) y = \sec (x) + C + \int - 1 d x + \int \sec^{2} (x) d x$

Schritt 7.5

Wende die Konstantenregel an.

$(\sec (x) + \tan (x)) y = \sec (x) + C - x + C + \int \sec^{2} (x) d x$

Schritt 7.6

Da die Ableitung von $\tan (x)$ gleich $\sec^{2} (x)$ ist, ist das Integral von $\sec^{2} (x)$ gleich $\tan (x)$ .

$(\sec (x) + \tan (x)) y = \sec (x) + C - x + C + \tan (x) + C$

Schritt 7.7

Vereinfache.

$(\sec (x) + \tan (x)) y = \sec (x) - x + \tan (x) + C$

Schritt 8

Teile jeden Ausdruck in $(\sec (x) + \tan (x)) y = \sec (x) - x + \tan (x) + C$ durch $\sec (x) + \tan (x)$ und vereinfache.

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Schritt 8.1

Teile jeden Ausdruck in $(\sec (x) + \tan (x)) y = \sec (x) - x + \tan (x) + C$ durch $\sec (x) + \tan (x)$ .

$\frac{(\sec (x) + \tan (x)) y}{\sec (x) + \tan (x)} = \frac{\sec (x)}{\sec (x) + \tan (x)} + \frac{- x}{\sec (x) + \tan (x)} + \frac{\tan (x)}{\sec (x) + \tan (x)} + \frac{C}{\sec (x) + \tan (x)}$

Schritt 8.2

Vereinfache die linke Seite.

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Schritt 8.2.1

Kürze den gemeinsamen Faktor von $\sec (x) + \tan (x)$ .

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Schritt 8.2.1.1

Kürze den gemeinsamen Faktor.

$\frac{(\sec (x) + \tan (x)) y}{\sec (x) + \tan (x)} = \frac{\sec (x)}{\sec (x) + \tan (x)} + \frac{- x}{\sec (x) + \tan (x)} + \frac{\tan (x)}{\sec (x) + \tan (x)} + \frac{C}{\sec (x) + \tan (x)}$

Schritt 8.2.1.2

Dividiere $y$ durch $1$ .

$y = \frac{\sec (x)}{\sec (x) + \tan (x)} + \frac{- x}{\sec (x) + \tan (x)} + \frac{\tan (x)}{\sec (x) + \tan (x)} + \frac{C}{\sec (x) + \tan (x)}$

Schritt 8.3

Vereinfache die rechte Seite.

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Schritt 8.3.1

Ziehe das Minuszeichen vor den Bruch.

$y = \frac{\sec (x)}{\sec (x) + \tan (x)} - \frac{x}{\sec (x) + \tan (x)} + \frac{\tan (x)}{\sec (x) + \tan (x)} + \frac{C}{\sec (x) + \tan (x)}$

Schritt 8.3.2

Vereinige die Zähler über dem gemeinsamen Nenner.

$y = \frac{\sec (x) - x}{\sec (x) + \tan (x)} + \frac{\tan (x)}{\sec (x) + \tan (x)} + \frac{C}{\sec (x) + \tan (x)}$

Schritt 8.3.3

Vereinige die Zähler über dem gemeinsamen Nenner.

$y = \frac{\sec (x) - x + \tan (x)}{\sec (x) + \tan (x)} + \frac{C}{\sec (x) + \tan (x)}$

Schritt 8.3.4

Vereinige die Zähler über dem gemeinsamen Nenner.

$y = \frac{\sec (x) - x + \tan (x) + C}{\sec (x) + \tan (x)}$