Solve y = e ^ - ∫ 2 wrt x(c+intPhi1*e^int2dxdx)

$$y = e ^ { - \int 2 d x } ( c + \int \Phi 1 \cdot e ^ { \int 2 d x } d x )$$

Solve for c

$$c=-\frac{\Phi _{1}e^{2x+С}}{2}+ye^{2x+С_{1}}-\frac{С_{2}}{2}$$

Steps for Solving Linear Equation

Use the distributive property to multiply e^{-\int 2\mathrm{d}x} by c+\int \Phi _{1}e^{\int 2\mathrm{d}x}\mathrm{d}x.

$$y=e^{-\int 2\mathrm{d}x}c+e^{-\int 2\mathrm{d}x}\int \Phi _{1}e^{\int 2\mathrm{d}x}\mathrm{d}x$$

Swap sides so that all variable terms are on the left hand side.

$$e^{-\int 2\mathrm{d}x}c+e^{-\int 2\mathrm{d}x}\int \Phi _{1}e^{\int 2\mathrm{d}x}\mathrm{d}x=y$$

Subtract e^{-\int 2\mathrm{d}x}\int \Phi _{1}e^{\int 2\mathrm{d}x}\mathrm{d}x from both sides.

$$e^{-\int 2\mathrm{d}x}c=y-e^{-\int 2\mathrm{d}x}\int \Phi _{1}e^{\int 2\mathrm{d}x}\mathrm{d}x$$

Reorder the terms.

$$ce^{-\int 2\mathrm{d}x}=-e^{-\int 2\mathrm{d}x}\int \Phi _{1}e^{\int 2\mathrm{d}x}\mathrm{d}x+y$$

The equation is in standard form.

$$\frac{1}{e^{2x+С}}c=-\frac{С}{e^{2x+С_{1}}}-\frac{\Phi _{1}}{2}+y$$

Divide both sides by e^{-\left(2x+С\right)}.

$$\frac{\frac{1}{e^{2x+С}}ce^{2x+С}}{1}=\frac{2ye^{2x+С_{1}}-\Phi _{1}e^{2x+С}-2С_{2}}{2e^{2x+С_{3}}\times \frac{1}{e^{2x+С}}}$$

Dividing by e^{-\left(2x+С\right)} undoes the multiplication by e^{-\left(2x+С\right)}.

$$c=\frac{2ye^{2x+С_{1}}-\Phi _{1}e^{2x+С}-2С_{2}}{2e^{2x+С_{3}}\times \frac{1}{e^{2x+С}}}$$

Divide \frac{2ye^{2x+С}-2С-e^{2x+С}\Phi _{1}}{2e^{2x+С}} by e^{-\left(2x+С\right)}.

$$c=-\frac{\Phi _{1}e^{2x+С}}{2}+ye^{2x+С_{1}}-С_{2}$$

Solve for y

$$y=Сe^{-2x-С_{1}}+ce^{-2x-С_{2}}+\frac{\Phi _{1}}{2}$$