Question

Find the general solution of the equation ${\displaystyle \frac{\text{d}y}{\text{d}x}-y}$ = 2x.

Solution: The equation ${\displaystyle \frac{\text{d}y}{\text{d}x}-y}$ = 2x

is of the form ${\displaystyle \frac{\text{d}y}{\text{d}x}+\text{P}y}$ = Q

where P = ${\displaystyle \square }$ and Q = ${\displaystyle \square }$

∴ I.F. = ${\displaystyle {\text{e}}^{\int -\text{d}x}}$ = e^–x

∴ the solution of the linear differential equation is

ye^–x = ${\displaystyle \int 2x\cdot {\text{e}}^{-x} \text{d}x+\text{c}}$

∴ ye^–x  = ${\displaystyle 2\int x\cdot {\text{e}}^{-x} \text{d}x+\text{c}}$

= ${\displaystyle 2\{x\int {\text{e}}^{-x}\text{d}x-\int \square \text{d}x\cdot \frac{\text{d}}{\text{d}x}\square \text{d}x\}+\text{c}}$

= ${\displaystyle 2\{x\frac{{\text{e}}^{-x}}{-1}-\int \frac{{\text{e}}^{-x}}{-1}\cdot 1\text{d}x\}+\text{c}}$

∴ ye^–x = ${\displaystyle -2x\cdot {\text{e}}^{-x}+2\int {\text{e}}^{-x}\text{d}x+\text{c}}$

∴ e^–xy = ${\displaystyle -2x\cdot {\text{e}}^{-x}+2\square +\text{c}}$

∴ ${\displaystyle y+\square +\square }$ = ce^x is the required general solution of the given differential equation

Answer 1

The equation ${\displaystyle \frac{\text{d}y}{\text{d}x}-y}$ = 2x

is of the form ${\displaystyle \frac{\text{d}y}{\text{d}x}+\text{P}y}$ = Q

where P = – 1 and Q = 2x

∴ I.F. = ${\displaystyle {\text{e}}^{\int -\text{d}x}}$ = e^–x

∴ the solution of the linear differential equation is

ye^–x = ${\displaystyle \int 2x\cdot {\text{e}}^{-x} \text{d}x+\text{c}}$

∴ ye^–x  = ${\displaystyle 2\int x\cdot {\text{e}}^{-x} \text{d}x+\text{c}}$

= ${\displaystyle 2\{x\int {\text{e}}^{-x}\text{d}x-\int {\text{e}}^{-x} \text{d}x\cdot \frac{\text{d}}{\text{d}x}x\text{d}x\}+\text{c}}$

= ${\displaystyle 2\{x\frac{{\text{e}}^{-x}}{-1}-\int \frac{{\text{e}}^{-x}}{-1}\cdot 1\text{d}x\}+\text{c}}$

∴ ye^–x = ${\displaystyle -2x\cdot {\text{e}}^{-x}+2\int {\text{e}}^{-x}\text{d}x+\text{c}}$

∴ e^–xy = ${\displaystyle -2x\cdot {\text{e}}^{-x}+2\frac{{\text{e}}^{-x}}{-1}+\text{c}}$

∴ y + 2x + 2 = ce^x is the required general solution of the given differential equation