Question

Determine whether the given differential equation is exact and solve it;

$\mathit{t}\frac{\mathbf{d}\mathbf{Q}}{\mathbf{d}\mathbf{t}}\mathbf{+}\mathit{Q}\mathbf{=}{\mathit{t}}^{\mathbf{4}}\mathit{l}\mathit{n}\mathit{t}$

Short answer

We will solve the given differential equation

Step-by-step solution

Solve the given differential equation by substituting

Often the first step in solving the differential equation consists of transforming it into another differential equation by means of a substantial.

For example: Suppose we wish to transform the first order differential equation dy/dx=f(x,y) by the substitution y=g(x,u), where u is regarded as a function of the variable x.

Final Answer

We rewrite the equation as

$\frac{dQ}{dt}+\frac{1}{t}Q=t^3\ln t$

which is a linear equation and whose integrating factor is

$\begin{array}{l}{e}^{\int \frac{1}{t}dt}=e^{\ln t}=t\\ \end{array}$

Multiply the equation with integrating factor to get:

$t\frac{dQ}{dt}+Q=t^4\ln t$

Upon integration we get,

$\begin{array}{c}\int \frac{d}{dt}\left[tQ\right]=\int t^4\ln tdt\\ tQ=\ln t\int t^{4}dt-\int \left(\frac{d}{dt}\left(\ln t\int t^{4}dt\right)dt\right)\\ Apply\int uvdt\\ =\ln t\left(\frac{t^5}{5}\right)-\int \left(\frac{1}{t}.\frac{t^5}{5}\right)dt\\ =\frac{t^5\ln t}{5}-\frac{1}{5}\int t^{4}dt\\ =\frac{t^5\ln t}{5}-\frac{1}{5}\left(\frac{t^5}{5}\right)\\ =\frac{t^5\ln t}{5}-\left(\frac{t^5}{25}\right)\\ =\frac{t^5}{5}\left(\ln t-\frac{1}{5}\right)\end{array}$

Hence, the final answer is :$\frac{t^5}{5}\left(\ln t-\frac{1}{5}\right)$$\frac{t^5}{5}\left(\ln t-\frac{1}{5}\right)$