Solve d^2x/dx^2=4y+e^x

$$\frac { d ^ { 2 } x } { d x ^ { 2 } } = 4 y + e ^ { x }$$

Solve for y

$$y=-\frac{e^{x}}{4}$$

Steps for Solving Linear Equation

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

$$4y+e^{x}=\frac{\mathrm{d}(x)}{\mathrm{d}x^{2}}$$

Subtract e^{x} from both sides.

$$4y=\frac{\mathrm{d}(x)}{\mathrm{d}x^{2}}-e^{x}$$

The equation is in standard form.

$$4y=-e^{x}$$

Divide both sides by 4.

$$\frac{4y}{4}=-\frac{e^{x}}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$y=-\frac{e^{x}}{4}$$

Solve for x

$$x=\ln(-4y)$$

$$y<0$$