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  • Need Solution for R.D.Sharma Maths Class 12 Chapter 21 Differential Equations Exercise Revision Exercise Question 18 Maths Textbook Solution.

Need Solution for R.D.Sharma Maths Class 12 Chapter 21 Differential Equations Exercise Revision Exercise Question 18 Maths Textbook Solution.

Answers (1)

Answer: y=\frac{\cos ^{7} x}{7}-\frac{\cos ^{5} x}{5}+\frac{2(x+1)^{5 / 2}}{5}-\frac{2(x+1)^{8 / 2}}{3}+C

Hint: Apply integration by parts method.

Given:\frac{dy}{dx}=\sin ^{3}x\cos ^{4}x+x\sqrt{x+1}

Solution:\frac{dy}{dx}=\sin ^{3}x\cos ^{4}x+x\sqrt{x+1}

\begin{aligned} &\left.\Rightarrow \int d y=\int \sin ^{3} x \cos ^{4} x+x \sqrt{x+1}\right) d x \text { (integrating both sides) } \\ &\left.\Rightarrow \quad y=\int \sin ^{3} x \cos ^{4} x d x+\int x \sqrt{x+1}\right) d x \\ &\Rightarrow \quad y=I_{1}+I_{2} \end{aligned}

Now, I_{1}=\int \sin ^{3}x\cos ^{4}xdx

=\int \left ( 1-\cos ^{2}x \right )\left ( \cos ^{4}x \right )\left ( \sin x \right )dx\; \; \; \; \; \; \; \; \; \; \; \left ( \because \cos ^{2}x+\sin ^{2}x=1 \right )

Let t=\cos x

\begin{aligned} &\Rightarrow \quad d t=-\sin x d x(\text { diff. w.r.tx }) \\ &\Rightarrow I_{1}=-\int t^{4}\left(1-t^{2}\right) d t \\ &\left.\Rightarrow I_{1}=\int t^{4}\left(t^{2}-1\right) d t=\int t^{6}-t^{4}\right) d t \\ &\Rightarrow \quad l_{1}=\frac{t^{7}}{7}-\frac{t^{5}}{5}+c_{1} \\ &=\frac{\cos ^{7} x}{7}-\frac{\cos ^{5} x}{5}+c_{1} \\ &\left.l_{2}=\int x \sqrt{x+1}\right) d x \end{aligned}

Let t^{2}=x+1

        \begin{aligned} &\Rightarrow 2 t d t=d x \quad(\text { diff. } w . r \cdot t x) \\ &\Rightarrow \quad l_{2}=2 \int\left(t^{2}-1\right) \cdot t \cdot t d t \\ &\Rightarrow \quad l_{2}=2 \int t^{4}-t^{2} d t \\ &\Rightarrow \quad I_{2}=\frac{2 t^{5}}{5}-\frac{2 t^{3}}{3}+c_{2} \end{aligned}

\therefore y=I_{1}+I_{2}

\begin{aligned} &\Rightarrow \mathrm{y}=\frac{\cos ^{7} \mathrm{x}}{7}-\frac{\cos ^{5} \mathrm{x}}{5}+\mathrm{c}_{1}+\frac{2(\mathrm{x}+1)^{5 / 2}}{5}-\frac{2(\mathrm{x}+1)^{8 / 2}}{3}+\mathrm{C}_{2} \\ &\Rightarrow \mathrm{y}=\frac{\cos ^{7} \mathrm{x}}{7}-\frac{\cos ^{5} \mathrm{x}}{5}+\frac{2(\mathrm{x}+1)^{5 / 2}}{5}-\frac{2(\mathrm{x}+1)^{8 / 2}}{3}+\mathrm{c} \quad\left(\because \mathrm{c}_{1}+\mathrm{C}_{2}=\mathrm{C}\right) \end{aligned}

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