Solve xpartial^2y/partialx^2-(2x-1)partialy/partialx+cx-1Delta+y=0

$$x \frac { \partial ^ { 2 } y } { \partial x ^ { 2 } } - ( 2 x - 1 ) \frac { \partial y } { \partial x } + c x - 1 \Delta + y = 0$$

Solve for c

$$\left\{\begin{matrix}c=-\frac{y-\Delta }{x}\text{, }&x\neq 0\\c\in \mathrm{R}\text{, }&y=\Delta \text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2x-1 by \frac{\mathrm{d}(y)}{\mathrm{d}x}.

$$x\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}-\left(2x\frac{\mathrm{d}(y)}{\mathrm{d}x}-\frac{\mathrm{d}(y)}{\mathrm{d}x}\right)+cx-\Delta +y=0$$

To find the opposite of 2x\frac{\mathrm{d}(y)}{\mathrm{d}x}-\frac{\mathrm{d}(y)}{\mathrm{d}x}, find the opposite of each term.

$$x\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}-2x\frac{\mathrm{d}(y)}{\mathrm{d}x}+\frac{\mathrm{d}(y)}{\mathrm{d}x}+cx-\Delta +y=0$$

Subtract x\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}} from both sides. Anything subtracted from zero gives its negation.

$$-2x\frac{\mathrm{d}(y)}{\mathrm{d}x}+\frac{\mathrm{d}(y)}{\mathrm{d}x}+cx-\Delta +y=-x\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}$$

Add 2x\frac{\mathrm{d}(y)}{\mathrm{d}x} to both sides.

$$\frac{\mathrm{d}(y)}{\mathrm{d}x}+cx-\Delta +y=-x\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}+2x\frac{\mathrm{d}(y)}{\mathrm{d}x}$$

Subtract \frac{\mathrm{d}(y)}{\mathrm{d}x} from both sides.

$$cx-\Delta +y=-x\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}+2x\frac{\mathrm{d}(y)}{\mathrm{d}x}-\frac{\mathrm{d}(y)}{\mathrm{d}x}$$

Add \Delta to both sides.

$$cx+y=-x\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}+2x\frac{\mathrm{d}(y)}{\mathrm{d}x}-\frac{\mathrm{d}(y)}{\mathrm{d}x}+\Delta$$

Subtract y from both sides.

$$cx=-x\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}+2x\frac{\mathrm{d}(y)}{\mathrm{d}x}-\frac{\mathrm{d}(y)}{\mathrm{d}x}+\Delta -y$$

The equation is in standard form.

$$xc=\Delta -y$$

Divide both sides by x.

$$\frac{xc}{x}=\frac{\Delta -y}{x}$$

Dividing by x undoes the multiplication by x.

$$c=\frac{\Delta -y}{x}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{y-\Delta }{c}\text{, }&c\neq 0\\x\in \mathrm{R}\text{, }&y=\Delta \text{ and }c=0\end{matrix}\right.$$