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Answer on Differential Equations Question for soniya

Question #6012
solve the differential equation: dy/dx + y = (x+1)e^x
Expert's answer
So, we have an equation:
<img style="width: 154px; height: 56px;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANEAAABMCAIAAAB8oMUlAAADS0lEQVR4nO2c25mDIBBGqYuCqIdqaMZisg/qhngNGH8Qznkz+ZLFeBwGZxbzAtBiSg8AugPnQA3OgRqcAzU4B2pwDtTgHKjBOVCDc6AG50ANzoEanAM1OAdqcA7U4ByowTlQg3OgBudADc6BGpwDNTgHanAO1OAcqCnp3OCtMcYY64eCowA1hePc4K0xLpQdBGgp7FxwhLnuKOvc4C3KdYfcuTmJM8Y4/zmzBrdM76ZXmH2bQutccJFTo1DRkQvTi5Fiy2N4PkLnxgj39mfwdiOZW2R4wTH5tobOuZVi22vWz1en6ActIXPuS+U+A11wBLn2kDm3TMy2Z9ZX7OLgLUGuQQo5N61eN5WaA11wGNckRebW4Kz38xS6StnGQOcwrlXk69b/8Lb/7I3nI21TY18Ji9W2qc+5NteqdRWW81srPhLx5RPXL6nFucm0FkNc6VRhT/eM22C8SsEZ48J4MHiXfCdV4tyc6zUnXNFmrXcGveNWZvQdU/Hss6rEuatU2qDya+MSTjNEzwUOxMqz7lpFEudu5OdJXM5pHjuXMcbBW2vthVupM+ek7VK7idxyFF/PVTc4tydd1HM2j2zwzg9jxj2ldHmVop6cE7dL7VT3olFYP6RV+O5wbmOY8Ueit+MfalQyL4j35NxMUrvUHAf32fvwictjqEiLFHc4twp065/nx2mLyLmz63bO4gvzVZC1S504d1Ry/vyOzNOMv+LMuUUlPP5xTseYTI9xTtQudTZnZwSQ++NcnMbdoNvr9erUuZR2qfxIs9ut9X439Zren88pHmB36ZyoXerg+g3eupD+9O7+desq+OYUGk7o0zlRu9RB1endWjMtYr8aiuD53MdxcMZYF3Bui9SLIaqBriLZoiqeWvJLOc1lZrb9hzZuiyibuCfV7XS/ElkvQd2bY5TpdulyvxJtu1TpvpI9irVXdbRfScl2qeM1rJyy0bef/UqabZd6HDXtV3Kh+A0Popr9Sq4Vv+FBVLdfSV7xGx5EdfuVZFaF4DnUt19Jbf8jBb+m0v1KCHMNU9t+JZnFb3gQde1Xkl38hgdRy34lF4vf8CAa6SuBB4FzoAbnQA3OgRqcAzU4B2pwDtTgHKjBOVCDc6AG50DNHxHro/iLc7uDAAAAAElFTkSuQmCC" alt="">
At first, let’s solve a simpler equation:
<img style="width: 166px; height: 163px;" src="data:image/png;base64,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" alt="">
Now, we suppose that C is a function of x, and put the result, we achieved, into initial equation:
<img style="width: 471px; height: 106px;" src="data:image/png;base64,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" alt="">
Answer:
<img style="width: 218px; height: 49px;" src="data:image/png;base64,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" alt="">

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