JEE Mathematics Differential Equations MCQs Set 25

Answer: (c) \(b-a\le\dfrac\pi{24}\)

Explanation:
1. \(\dfrac{ff'}{f^4+1}\ge1\), so \(\dfrac12\tan^{-1}f^2\) increases at rate at least \(1\).
2. Integrating: \(b-a\le\dfrac12\left(\dfrac\pi3-\dfrac\pi4\right)=\dfrac\pi{24}\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (d) \(f\left(\dfrac12\right)<\dfrac12\) and \(f\left(\dfrac13\right)<\dfrac13\)

Explanation:
1. Differentiate: \(\sqrt{1-f'^2}=f\), so \(f'^2=1-f^2\) and, with \(f(0)=0\), \(f=\sin x\).
2. \(\sin\dfrac12\approx0.479<\dfrac12\) and \(\sin\dfrac13\approx0.327<\dfrac13\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

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

Explanation:
1. Let \(u=x^2y^2\): \(u'=2xy^2+2x^2yy'\), so the equation becomes \(u'=\tan u\).
2. \(\dfrac{\cos u}{\sin u}du=dx\) gives \(\sin u=ce^x\). At \(x=1\), \(u=\dfrac\pi2\), so \(c=e^{-1}\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (b) \(\sec\dfrac yx=cxy\)

Explanation:
1. Put \(y=vx\): the equation becomes \(2vx^2\cos v\,dx=x^3(v\sin v-\cos v)\,dv\).
2. So \(\dfrac{2\,dx}x=\left(\tan v-\dfrac1v\right)dv\), giving \(x^2v\cos v=c\), i.e. \(xy\cos\dfrac yx=c\) and \(\sec\dfrac yx=c\,xy\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (a) \(e^{-2x}y^2+2\ln|y|=c\)

Explanation:
1. Treat \(x\) as a function of \(y\): \(\dfrac{dx}{dy}=\dfrac{e^{2x}+y^2}{y^3}\). Let \(z=e^{-2x}\): \(\dfrac{dz}{dy}+\dfrac{2z}y=-\dfrac2{y^3}\).
2. Integrating factor \(y^2\): \(\dfrac d{dy}(zy^2)=-\dfrac2y\), so \(zy^2=-2\ln|y|+c\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (b) \(ny^2+x^2=\text{constant}\)

Explanation:
1. \(y=\dfrac{x^n}{a^{n-1}}\) gives \(\dfrac{dy}{dx}=\dfrac{ny}x\) (eliminating \(a\)).
2. For orthogonal curves \(\dfrac{dy}{dx}=-\dfrac x{ny}\), i.e. \(ny\,dy+x\,dx=0\), so \(ny^2+x^2=c\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (d) \(2^{\sin x}+c\,2^x\)

Explanation:
1. The integrating factor is \(e^{-x\ln2}=2^{-x}\).
2. \(\dfrac d{dx}\left(y2^{-x}\right)=2^{\sin x-x}(\cos x-1)\ln2=\dfrac d{dx}2^{\sin x-x}\), so \(y2^{-x}=2^{\sin x-x}+c\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (a) \(t=\dfrac{g(x)}{x+c}\)

Explanation:
1. This is a Bernoulli equation; put \(u=\dfrac1t\): \(u'=-\dfrac{t'}{t^2}=-\dfrac{g'}g\,u+\dfrac1g\).
2. So \((ug)'=1\), \(ug=x+c\) and \(t=\dfrac g{x+c}\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (d) \(\dfrac1{3x}+\dfrac{2x^2}3\)

Explanation:
1. By L'Hopital in \(t\): \(2xf(x)-x^2f'(x)=1\), i.e. \(f'-\dfrac{2f}x=-\dfrac1{x^2}\).
2. Integrating factor \(x^{-2}\): \(\left(\dfrac f{x^2}\right)'=-\dfrac1{x^4}\), so \(f=\dfrac1{3x}+Cx^2\); \(f(1)=1\) gives \(C=\dfrac23\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (a) \(2\)

Explanation:
1. Write \(\dfrac{dx}{dy}=xy(x^2\sin y^2+1)\) and let \(z=\dfrac1{x^2}\): \(\dfrac{dz}{dy}+2yz=-2y\sin y^2\).
2. With factor \(e^{y^2}\): \(z=\dfrac{\cos y^2-\sin y^2}2+C'e^{-y^2}\), i.e. \(x^2\left(\cos y^2-\sin y^2+2C'e^{-y^2}\right)=2\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (b) \(8\)

Explanation:
1. Let \(A=\int_0^xy\), \(B=\int_0^xty\). Differentiating the relation gives \(A-B=x^2y\), while the relation gives \(A-B=\dfrac Bx\); so \(B=x^3y\).
2. Differentiating \(B\): \(xy=3x^2y+x^3y'\), so \(\dfrac{y'}y=\dfrac1{x^2}-\dfrac3x\) and \(y=\dfrac{Ce^{-1/x}}{x^3}\) with \(C=e^2\).
3. \(y\left(\dfrac12\right)=e^2e^{-2}\cdot8=8\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (c) \(7\)

Explanation:
1. Rearranging: \(x^2f'(x)\left(f\left(\frac1x\right)-1\right)=f'\left(\frac1x\right)\left(f(x)-1\right)\), which is satisfied by \(f(x)=x^n+1\).
2. \(f(1)=2\) holds for any \(n\) and \(f(5)=26=5^2+1\) gives \(n=2\); so \(f(6)=37\) and \(f(6)-30=7\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (d) \(1\)

Explanation:
1. The series is that of \(f(x)=\dfrac{\sin^{-1}x}{\sqrt{1-x^2}}\).
2. \(A=\displaystyle\int_{1/2}^{\sqrt3/2}\dfrac{\sin^{-1}x}{\sqrt{1-x^2}}dx=\dfrac12\left[\left(\sin^{-1}x\right)^2\right]=\dfrac12\left(\dfrac{\pi^2}9-\dfrac{\pi^2}{36}\right)=\dfrac{\pi^2}{24}\).
3. \(4A=\dfrac{\pi^2}6\approx1.645\), so \([4A]=1\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (c) The order of the differential equation \(\sqrt{1+\dfrac{d^2y}{dx^2}}=x\) is \(1\)

Explanation:
1. Squaring gives \(1+y''=x^2\), which contains the second derivative: order \(2\), so the first statement is incorrect.
2. The others check: with \(y=vx\) one gets \(v+\sqrt{1+v^2}=cx\); the family has auxiliary roots \(1\pm i\), giving \(y''-2y'+2y=0\); and the linear equation in \(x\) has factor \(e^{\tan^{-1}y}\), giving the stated solution.

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (b) \(\sqrt{x^2+y^2}=a\sin\left(\tan^{-1}\dfrac yx+c\right)\)

Explanation:
1. In polar form \(x\,dx+y\,dy=r\,dr\) and \(x\,dy-y\,dx=r^2d\theta\), so \(\dfrac{dr}{r\,d\theta}=\dfrac{\sqrt{a^2-r^2}}r\).
2. \(\dfrac{dr}{\sqrt{a^2-r^2}}=d\theta\) gives \(\sin^{-1}\dfrac ra=\theta+c\), i.e. \(r=a\sin(\theta+c)\) with \(\theta=\tan^{-1}\dfrac yx\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (a) I, II and III only

Explanation:
1. I has degree \(-1\), II has degree \(-\dfrac13\) and III has degree \(1\) (the logarithm difference is of degree \(0\)).
2. In IV the first part is of degree \(1\) but \(y^2\tan(\cdot)\) is of degree \(2\), so it is not homogeneous.

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (a) \(6\)

Explanation:
1. The function is a general solution of the linear equation with auxiliary equation \(m^3-7m+6=0\).
2. That equation is \(y'''-7y'+6y=0\), so \(k=6\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (b) \(2\)

Explanation:
1. The highest-order derivative is \(\dfrac{d^3y}{dx^3}\) (order \(3\)).
2. Its highest power is \(2\), so the degree is \(2\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (d) \(5\)

Explanation:
1. \(y=x^3-2x^2+x+c_2\) and \(y(1)=5\) gives \(c_2=5\).
2. Values: \(y(0)=5\), \(y\left(\dfrac13\right)\approx5.15\), \(y(1)=5\), \(y(2)=7\). The minimum is \(5\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (a) \(y(1+x^2)=\dfrac43x^3+c\)

Explanation:
1. The left side \((1+x^2)y'+2xy\) is \(\dfrac d{dx}\left[(1+x^2)y\right]\).
2. So \((1+x^2)y=\displaystyle\int4x^2dx=\dfrac43x^3+c\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (c) \(y(1+x)=x+ce^x\)

Explanation:
1. With factor \(e^{-x}(1+x)\): \(\dfrac d{dx}\left[ye^{-x}(1+x)\right]=e^{-x}(1-x)=\dfrac d{dx}\left(xe^{-x}\right)\).
2. So \(y(1+x)e^{-x}=xe^{-x}+c\), i.e. \(y(1+x)=x+ce^x\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (a) \(e^{x/\sqrt{1-x^2}}\)

Explanation:
1. \(\displaystyle\int\dfrac{dx}{(1-x^2)^{3/2}}=\dfrac x{\sqrt{1-x^2}}\) (put \(x=\sin\theta\): \(\int\sec^2\theta\,d\theta\)).
2. The integrating factor is \(e^{x/\sqrt{1-x^2}}\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.

Answer: (b) \(y=\dfrac x{\sqrt{1-x^2}}+ce^{-x/\sqrt{1-x^2}}\)

Explanation:
1. Let \(u=\dfrac x{\sqrt{1-x^2}}\); then \(u'=(1-x^2)^{-3/2}\) and \(u'+\dfrac u{(1-x^2)^{3/2}}=\dfrac{\sqrt{1-x^2}+x}{(1-x^2)^2}\), so \(u\) is a particular solution.
2. The homogeneous part gives \(ce^{-u}\), so the general solution is \(y=u+ce^{-u}\).

Teacher's Note:
1. Identify the type first (variable separable, homogeneous, linear or exact) and then apply the standard method; verify by differentiating the answer.
2. Remember that order is the highest derivative present and degree is its power after clearing radicals.