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  • Please solve RD Sharma maths Class 12 Chapter 21 Differential Equation Exercise Revision Exercise (RE) Question 70 textbook solution.

Please solve RD Sharma maths Class 12 Chapter 21 Differential Equation Exercise Revision Exercise (RE) Question 70 textbook solution.

Answers (1)

Answer : 2 y-1=e^{x}(\sin x-\cos x)

Hint               : using integration by parts

Given :    \frac{d y}{d x}=e^{x} \sin x

Solution : \frac{d y}{d x}=e^{x} \sin x

               d y=e^{x} \sin x d x

Integrating both sides

\begin{aligned} &\int d y=\int e^{x} \sin x d x\\ &y=\int e^{x} \sin x d x \end{aligned}               ....(i)

Using integration by parts

Let          \begin{aligned} &I=\int e^{x} \sin x d x=\sin x e^{x}-\int \cos x e^{x} d x \\ \end{aligned}

                   \begin{aligned} &\qquad=\sin x e^{x}-\left[\cos x e^{x}-\int(-\sin x) e^{x} d x\right] \\ \end{aligned}

              \begin{aligned} &I=\sin x e^{x}-\cos x e^{x}-\int \sin x e^{x} d x \\ \end{aligned}

             \begin{aligned} &I=\sin x e^{x}-\cos x e^{x}-I \\ \end{aligned}

         \begin{aligned} &2 I=\sin x e^{x}-\cos x e^{x} \\ \end{aligned}

          \begin{aligned} &I=\frac{e^{x}(\sin x-\cos x)}{2} \end{aligned}

Put in (i), we get,

\begin{gathered} y=\frac{e^{x}(\sin x-\cos x)}{2} \\ 2 y=e^{x}(\sin x-\cos x) \\ \qquad \begin{array}{r} y=\frac{1}{2} e^{x}(\sin x-\cos x)+c \end{array} \end{gathered}                          ....(ii)

Given curve passes through (0,0)

Putting x = 0 , y = 0 in equation

Therefore ,          \begin{aligned} &0=\frac{1}{2} e^{0}(\sin 0-\cos 0)+c \\ \end{aligned}

                            \begin{aligned} &0=\frac{1}{2}(1)(0-1)+c \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[e^{0}=1\right] \\ \end{aligned}

                            \begin{aligned} &0=\frac{-1}{2}+c \end{aligned}

                          c=\frac{1}{2}

Putting value of C in (ii)

            \begin{gathered} y=\frac{1}{2} e^{x}(\sin x-\cos x)+\frac{1}{2} \\ y-\frac{1}{2}=\frac{1}{2} e^{x}(\sin x-\cos x) \\ \frac{2 y-1}{2}=\frac{e^{x}}{2}(\sin x-\cos x) \\ 2 y-1=e^{x}(\sin x-\cos x) \end{gathered}

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