Solve d2y/dx^2-4y=xsinhx

$$\frac { d 2 y } { d x ^ { 2 } } - 4 y = x \sinh x$$

Solve for d

$$\left\{\begin{matrix}d=\frac{d_{2}y}{x^{2}\left(x\sinh(x)+4y\right)}\text{, }&y\neq 0\text{ and }d_{2}\neq 0\text{ and }x\neq 0\text{ and }y\neq -\frac{x\sinh(x)}{4}\\d\neq 0\text{, }&y=-\frac{x\sinh(x)}{4}\text{ and }d_{2}=0\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable d cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by dx^{2}.

$$d_{2}y-4ydx^{2}=x\sinh(x)dx^{2}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$d_{2}y-4ydx^{2}=x^{3}\sinh(x)d$$

Subtract x^{3}\sinh(x)d from both sides.

$$d_{2}y-4ydx^{2}-x^{3}\sinh(x)d=0$$

Subtract d_{2}y from both sides. Anything subtracted from zero gives its negation.

$$-4ydx^{2}-x^{3}\sinh(x)d=-d_{2}y$$

Combine all terms containing d.

$$\left(-4yx^{2}-x^{3}\sinh(x)\right)d=-d_{2}y$$

The equation is in standard form.

$$\left(-x^{3}\sinh(x)-4yx^{2}\right)d=-d_{2}y$$

Divide both sides by -4x^{2}y-\sinh(x)x^{3}.

$$\frac{\left(-x^{3}\sinh(x)-4yx^{2}\right)d}{-x^{3}\sinh(x)-4yx^{2}}=-\frac{d_{2}y}{-x^{3}\sinh(x)-4yx^{2}}$$

Dividing by -4x^{2}y-\sinh(x)x^{3} undoes the multiplication by -4x^{2}y-\sinh(x)x^{3}.

$$d=-\frac{d_{2}y}{-x^{3}\sinh(x)-4yx^{2}}$$

Divide -d_{2}y by -4x^{2}y-\sinh(x)x^{3}.

$$d=\frac{d_{2}y}{x^{2}\left(x\sinh(x)+4y\right)}$$

Variable d cannot be equal to 0.

$$d=\frac{d_{2}y}{x^{2}\left(x\sinh(x)+4y\right)}\text{, }d\neq 0$$

Solve for d_2

$$d_{2}=\frac{dx^{2}\left(x\sinh(x)+4y\right)}{y}$$

$$y\neq 0\text{ and }d\neq 0\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by dx^{2}.

$$d_{2}y-4ydx^{2}=x\sinh(x)dx^{2}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$d_{2}y-4ydx^{2}=x^{3}\sinh(x)d$$

Add 4ydx^{2} to both sides.

$$d_{2}y=x^{3}\sinh(x)d+4ydx^{2}$$

The equation is in standard form.

$$yd_{2}=dx^{3}\sinh(x)+4dyx^{2}$$

Divide both sides by y.

$$\frac{yd_{2}}{y}=\frac{dx^{2}\left(x\sinh(x)+4y\right)}{y}$$

Dividing by y undoes the multiplication by y.

$$d_{2}=\frac{dx^{2}\left(x\sinh(x)+4y\right)}{y}$$