First slide

Methods of solving first order first degree differential equations

Question

$The general solution of the D.E y^{1} + y φ^{1} (x) - φ (x) φ^{1} (x) = 0 where φ (x) is a known function in$

Moderate

A

${y=ce}^{-φ(x)} +φ(x)-1$
B

${y=ce}^{φ(x)} +φ(x)-1$
C

${y=ce}^{-φ(x)} -φ(x)+1$
D

${y=ce}^{-φ(x)} +φ(x)+1$

Solution

$\begin{array}{l} \frac{dy}{dx} + y φ^{1} (x) = φ (x) φ^{1} (x) \\ I \cdot F = e^{\int φ^{'} (x) d x} = e^{φ (x)} \\ Solution of the D.E. {ye}^{φ (x)} = \int e^{φ (x)} \cdot φ (x) φ^{1} (x) dx \\ {ye}^{φ (x)} = \int e^{t} \cdot tdt = e^{t} (t - 1) + c \\ {ye}^{φ (x)} = e^{φ (x)} (φ (x) - 1) + c \\ y = {ce}^{- φ (x)} + φ (x) - 1 \end{array}$

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