NCERT Exemplar Solutions

Q-75. The solution of the differential equation$\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{2\mathrm{xy}}{1+{\mathrm{x}}^2}=\frac{1}{{\left(1+{\mathrm{x}}^2\right)}^2}$ is:

$$\begin{gathered} \left(\mathrm{A}\right) \mathrm{y}(1+{\mathrm{x}}^2)=\mathrm{c}+{\tan }^{–1 }\mathrm{x} \left(\mathrm{B}\right) \frac{\mathrm{y}}{1+{\mathrm{x}}^2}=\mathrm{c}+{\tan }^{-1}\mathrm{x} \\ \left(\mathrm{C}\right) \mathrm{y} \log (1 + {\mathrm{x}}^2)=\mathrm{c}+{\tan }^{–1}\mathrm{x} \left(\mathrm{D}\right) \mathrm{y}(1+{\mathrm{x}}^2)=\mathrm{c}+{\sin }^{–1}\mathrm{x} \end{gathered}$$

74. The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}={\mathrm{e}}^{\mathrm{x}-\mathrm{y}}+{\mathrm{x}}^2 {\mathrm{e}}^{-\mathrm{y}}$ is:

$$\begin{gathered} \left(\mathrm{A}\right) \mathrm{y}={\mathrm{e}}^{\mathrm{x}-\mathrm{y}}-{\mathrm{x}}^2 {\mathrm{e}}^{-\mathrm{y}}+\mathrm{c} \left(\mathrm{B}\right) {\mathrm{e}}^{\mathrm{y}}-{\mathrm{e}}^{\mathrm{x}}=\frac{{\mathrm{x}}^3}{3}+\mathrm{c} \\ \left(\mathrm{C}\right) {\mathrm{e}}^{\mathrm{y}}+{\mathrm{e}}^{\mathrm{x}}=\frac{{\mathrm{x}}^3}{3}+\mathrm{c} \left(\mathrm{D}\right) {\mathrm{e}}^{\mathrm{x}}-{\mathrm{e}}^{\mathrm{y}}=\frac{{\mathrm{x}}^3}{3}+\mathrm{c} \end{gathered}$$

Q- 73. The general solution of the differential equation (e^x + 1) ydy = (y + 1) e^x dx  is:

(A) (y + 1) = k (e^x + 1)                      (B) y + 1 = e^x + 1 + k

(C) y = log {k (y + 1) (e^x + 1)}          (D) $\mathrm{y}=\log \left\{\frac{\left({\mathrm{e}}^{\mathrm{x}}+1\right)}{\mathrm{y}+1}\right\}+\mathrm{k}$

Q-72. Solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{\mathrm{y}}{\mathrm{x}}=\sin \mathrm{x}$is :

(A) x (y + cosx) = sinx + c                  (B) x (y – cosx) = sinx + c

(C) xy cosx = sinx + c                         (D) x (y + cosx) = cosx + c

Q-71. General solution of tan sec$\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y} \tan \mathrm{x}=\sec \mathrm{x}$ is :

(A) y secx = tanx + c             (B) y tanx = secx + c             (C) tanx = y tanx + c (D) x secx = tany + c

Q-70. Which of the following is the general solution of 2 2 2 0d y dy y dx dx − + =?

(A) y = (Ax + B)e^x      (B) y = (Ax + B)e^–x     (C) y = Ae x + Be^–x     (D) y = Acosx + Bsinx