Question

Solve the differential equation
\[\dfrac{dy}{dx}-3y\cot (x)=\sin (2x)\], given \[y=2\], when \[x=\dfrac{\pi }{2}\].

Answer 1

Hint: We will begin the solution by comparing the given equation with the general form, \[\dfrac{dy}{dx}+Py=Q\] and find the values of P and Q. Once we have determined the values of P and Q, then we will be using the formula for the Integrating factor, which is given by, \[\text{IF}={{e}^{\int{Pdx}}}\] and substitute the value of P to calculate it. In the end, when we have calculated or determined P, Q, and the integrating factor, we will be using the formula of the solution of the differential equation, given by, \[y\times \text{IF}=\int{Q\cdot \text{IF}dx}\] then substitute the value of given x and y to find constant \[c\] and substituting in the obtained equation then you will get the required solution.

Complete step-by-step solution:
Let us consider the given equation,
\[\dfrac{dy}{dx}-3y\cot (x)=\sin (2x)\]
This equation can be compared with the standard form, \[\dfrac{dy}{dx}+Py=Q\] to find the value of P and Q.
Thus, on comparing we get;
\[p=-3\cot (x)\] and, \[Q=\sin (2x)\]
Now, we will now be calculating the integrating factor, using the formula;
\[\text{IF}={{e}^{\int{Pdx}}}\]
We will now substitute \[p=-3\cot (x)\] into the above formula, and find the value of the integrating factor.
\[\Rightarrow {{e}^{\int{-3\cot (x)dx}}}={{e}^{-3\int{\cot (x)dx}}}\]
By using the formula of integration that is \[\int{\cot (x)dx}=\left| \sin (x) \right|\] we get:
\[\Rightarrow {{e}^{\int{-3\cot (x)dx}}}={{e}^{-3\log \left| \sin (x) \right|}}\]
By using the property of logarithmic that is \[n\log m=\log {{m}^{n}}\]we get:
\[\Rightarrow {{e}^{\int{-3\cot (x)dx}}}={{e}^{\log {{\left| \sin (x) \right|}^{-3}}}}\]
Above equation can also be simplified as
\[\Rightarrow {{e}^{\int{-3\cot (x)dx}}}={{e}^{\log \dfrac{1}{\left| {{\sin }^{3}}(x) \right|}}}\]
By using the property of trigonometry that is \[\sin (x)=\dfrac{1}{\text{cosec}(x)}\] we get:
\[\Rightarrow {{e}^{\int{-3\cot (x)dx}}}={{e}^{\log \left| \text{cose}{{\text{c}}^{3}}(x) \right|}}\]
By using the property of logarithmic that is \[{{e}^{\log x}}=x\]
\[\Rightarrow {{e}^{\int{-3\cot (x)dx}}}=\text{cose}{{\text{c}}^{3}}(x)\]
Thus, we get the value of the integrating factor to be equal to \[\text{cose}{{\text{c}}^{3}}(x)\]
Now, we know that the solution to the differential equation of the given form, is given by;
\[y\times \text{IF}=\int{Q\cdot \text{IF}dx}\]. Thus, we substitute the given values into the formula to find the solution to the given equation.
\[\Rightarrow y\times \text{cose}{{\text{c}}^{3}}(x)=\int{\sin (2x)\cdot \text{cose}{{\text{c}}^{3}}(x)dx}\]
By using the property of half angle formula that means \[\sin (2x)=2 \sin (x)\cos (x)\] and also use the property of trigonometry that means \[\text{cosec}(x)=\dfrac{1}{\sin (x)}\] we get:
\[\Rightarrow y\times \text{cose}{{\text{c}}^{3}}(x)=\int{\dfrac{2\sin (x)\cos (x)}{{{\sin }^{3}}(x)}dx}\]
By simplifying further, we get:
\[\Rightarrow y\times \text{cose}{{\text{c}}^{3}}(x)=\int{\dfrac{2\cos (x)}{{{\sin }^{2}}(x)}dx}\]
By further solving this above equation we get:
\[\Rightarrow y\text{cose}{{\text{c}}^{3}}(x)=\int{\dfrac{2\cos (x)}{\sin (x)}\times \dfrac{1}{\sin (x)}dx}\]
By using the property of trigonometry, that means \[\cot (x)=\dfrac{\cos (x)}{\sin (x)}\] and \[\text{cosec}(x)=\dfrac{1}{\sin (x)}\] we get:
\[\Rightarrow y\text{cose}{{\text{c}}^{3}}(x)=2\int{\cot (x)\text{cosec}(x)dx}\]
Now, we will use the know integration that is \[\int{\cot (x)\text{cosec}(x)=-\text{cosec}(x)+c}\] we get:
\[\Rightarrow y\text{cose}{{\text{c}}^{3}}(x)=2(-\text{cosec}(x)) + c\]
Divide \[\text{cose}{{\text{c}}^{3}}(x)\]on both sides we get:
\[\Rightarrow y=\dfrac{-2\text{cosec}(x)}{\text{cose}{{\text{c}}^{3}}(x)}+\dfrac{c}{\text{cose}{{\text{c}}^{3}}(x)}\]
By simplifying further, we get:
\[\Rightarrow y=\dfrac{-2}{\text{cose}{{\text{c}}^{2}}(x)}+\dfrac{c}{\text{cose}{{\text{c}}^{3}}(x)}\]
Again, using the property of trigonometry that is \[\text{cosec}(x)=\dfrac{1}{\sin (x)}\] we get:
\[\Rightarrow y=-2{{\sin }^{2}}(x)+c\; {{\sin }^{3}}(x)--(1)\]
Now, in the question value is given, that is \[y=2\], when \[x=\dfrac{\pi }{2}\].
Substituting these values on above equation we get:
\[\Rightarrow 2=-2{{\sin }^{2}}\left( \dfrac{\pi }{2} \right)+c\; {{\sin }^{3}}\left( \dfrac{\pi }{2} \right)--(2)\]
As we know that \[\sin \left( \dfrac{\pi }{2} \right)=1\] after substituting the above equation (2) we get:
\[\Rightarrow 2=-\left( 2\times 1 \right)+\left( c\; \times 1 \right)\]
By simplifying further, we get:
\[\Rightarrow 2=-2+c\]
Therefore, we get:
\[\therefore c= 4\]
By substituting the above value on equation (1) we get:
\[\Rightarrow y=-2{{\sin }^{2}}(x)+4{{\sin }^{3}}(x)\]
So, this is a required solution.

Note: There are a variety of approaches to solving these types of problems, such as starting with trigonometric relations and equations and then isolating the variables to get the answer to the given equation. However, after a certain number of steps, you'll see that the equation's complexity has increased. So, wherever you find equations having the term, \[\dfrac{dy}{dx}\] and the question is asking to find its solutions, just use the method of integrating factor. This method is time saving and reduces the complexity of the given equation.