Issolvi ∫ y=e^x(acosx+bsinx)

$$\int y = e ^ { x } ( a \cos x + b \sin x )$$

Solvi għal a

$$\left\{\begin{matrix}a=\frac{-b\sin(x)e^{x}+xy+С}{\cos(x)e^{x}}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}+\frac{\pi }{2}\\a\in \mathrm{R}\text{, }&С=b\sin(x)e^{x}-xy\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }x=\frac{\pi \left(2n_{1}+1\right)}{2}\end{matrix}\right.$$

Passi biex Issolvi l-Ekwazzjoni Lineari

Uża l-propjetà distributtiva biex timmultiplika e^{x} b'a\cos(x)+b\sin(x).

$$\int y\mathrm{d}x=e^{x}a\cos(x)+e^{x}b\sin(x)$$

Ibdel in-naħat sabiex it-termini varjabbli kollha jkunu fuq in-naħa tax-xellug.

$$e^{x}a\cos(x)+e^{x}b\sin(x)=\int y\mathrm{d}x$$

Naqqas e^{x}b\sin(x) miż-żewġ naħat.

$$e^{x}a\cos(x)=\int y\mathrm{d}x-e^{x}b\sin(x)$$

L-ekwazzjoni hija f'forma standard.

$$\cos(x)e^{x}a=-b\sin(x)e^{x}+xy+С$$

Iddividi ż-żewġ naħat b'e^{x}\cos(x).

$$\frac{\cos(x)e^{x}a}{\cos(x)e^{x}}=\frac{-b\sin(x)e^{x}+xy+С}{\cos(x)e^{x}}$$

Meta tiddividi b'e^{x}\cos(x) titneħħa l-multiplikazzjoni b'e^{x}\cos(x).

$$a=\frac{-b\sin(x)e^{x}+xy+С}{\cos(x)e^{x}}$$

Iddividi yx+С-e^{x}b\sin(x) b'e^{x}\cos(x).

$$a=\frac{\frac{xy+С}{e^{x}}-b\sin(x)}{\cos(x)}$$

Solvi għal b

$$\left\{\begin{matrix}b=\frac{-a\cos(x)e^{x}+xy+С}{\sin(x)e^{x}}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}\\b\in \mathrm{R}\text{, }&С=a\cos(x)e^{x}-xy\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }x=\pi n_{1}\end{matrix}\right.$$

Passi biex Issolvi l-Ekwazzjoni Lineari

Uża l-propjetà distributtiva biex timmultiplika e^{x} b'a\cos(x)+b\sin(x).

$$\int y\mathrm{d}x=e^{x}a\cos(x)+e^{x}b\sin(x)$$

Ibdel in-naħat sabiex it-termini varjabbli kollha jkunu fuq in-naħa tax-xellug.

$$e^{x}a\cos(x)+e^{x}b\sin(x)=\int y\mathrm{d}x$$

Naqqas e^{x}a\cos(x) miż-żewġ naħat.

$$e^{x}b\sin(x)=\int y\mathrm{d}x-e^{x}a\cos(x)$$

L-ekwazzjoni hija f'forma standard.

$$\sin(x)e^{x}b=-a\cos(x)e^{x}+xy+С$$

Iddividi ż-żewġ naħat b'e^{x}\sin(x).

$$\frac{\sin(x)e^{x}b}{\sin(x)e^{x}}=\frac{-a\cos(x)e^{x}+xy+С}{\sin(x)e^{x}}$$

Meta tiddividi b'e^{x}\sin(x) titneħħa l-multiplikazzjoni b'e^{x}\sin(x).

$$b=\frac{-a\cos(x)e^{x}+xy+С}{\sin(x)e^{x}}$$

Iddividi yx+С-e^{x}a\cos(x) b'e^{x}\sin(x).

$$b=\frac{\frac{xy+С}{e^{x}}-a\cos(x)}{\sin(x)}$$