Question

Differential Equation Potpourri For each of the following differential equations, find at least one particular solution. You will need to call on past experience with functions you have differentiated. For a greater challenge, find the general solution. (a) \(y^{\prime}=x\) (b)\(y^{\prime}=-x\) (c)\(y^{\prime}=y\) (d)\(y^{\prime}=-y\) (e)\(y^{\prime \prime}=-y\)

Short answer

The solutions of the given differential equations are: (a) \(\frac{1}{2}x^2 + C\) (b) \(-\frac{1}{2}x^2 + C\) (c) \(e^{x}\) (d) \(e^{-x}\) (e) \(C_1*cos(x) +C_2*sin(x)\)

Step-by-step solution

Identify Type and Solve Equation (a)

Equation (a) is a first-order linear differential equation. Integrate both sides to find the particular solution. The integral of \(x\) with respect to \(x\) is \(\frac{1}{2}x^2\). The general solution is \(\frac{1}{2}x^2 + C\) where C is constant.

Identify Type and Solve Equation (b)

Equation (b) is a first-order linear differential equation. Integrate both sides to find the particular solution. The integral of \(-x\) with respect to \(x\) is \(-\frac{1}{2}x^2\). The general solution is \(-\frac{1}{2}x^2 + C\) where C is a constant.

Identify Type and Solve Equation (c)

Equation (c) is a first-order linear differential equation. The solution of this type of equation is obtained by separating variables. Following the separation of variables method and taking integral on both sides, the solution will be \(y= e^x\).

Identify Type and Solve Equation (d)

Equation (d) is a first-order linear differential equation. The solution of this type of equation is obtained by separating variables. Following the separation of variables method and taking integral on both sides, the solution will be \(y= e^{-x}\).

Identify Type and Solve Equation (e)

Equation (e) is a second-order linear differential equation. This can be solved using methods like undetermined coefficients or variation of parameters. However, since the differential equation is simple, we can guess that sine and cosine should be solutions because their second derivatives are themselves with a minus sign. So the general solution is \(y=C_1*cos(x) +C_2*sin(x)\).